Probability Worksheet 4, problem 7: Sock drawer experiment with data on white, blue, black, and tan socks.

Let’s solve this step by step.

We are told that you randomly picked 20 socks from your drawer, and here’s what you got:

- White: 12
- Blue: 1
- Black: 3
- Tan: 4

Total = 12 + 1 + 3 + 4 = 20 socks — that checks out.

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Part A: Find the experimental probability of each color.

Experimental probability = (number of times event happened) ÷ (total number of trials)

So for each color:

- P(white) = 12 ÷ 20 = 12/20 → simplify to 3/5 or 0.6
- P(blue) = 1 ÷ 20 = 1/20 or 0.05
- P(black) = 3 ÷ 20 = 3/20 or 0.15
- P(tan) = 4 ÷ 20 = 4/20 → simplify to 1/5 or 0.2

You can leave as fractions or decimals — both are fine. We’ll use fractions since they’re exact.

✔ So:
P(white) = 12/20 = 3/5
P(blue) = 1/20
P(black) = 3/20
P(tan) = 4/20 = 1/5

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Part B: Based on your experiment, how many socks of each color are in your drawer?

You know there are 64 individual socks total in the drawer (since 32 pairs × 2 = 64).

Use the probabilities from Part A to estimate how many of each color are in the full drawer.

Multiply each probability by 64:

- White: (12/20) × 64 = (3/5) × 64 = (3 × 64) ÷ 5 = 192 ÷ 5 = 38.4 → but we can’t have 0.4 sock! Hmm… let’s keep it as fraction for now or round later? Actually, since we’re predicting based on sample, maybe we should use the ratio directly.

Wait — better way: The sample was 20 socks. Full drawer is 64 socks.

So scale up using ratio: 64 ÷ 20 = 3.2

That means multiply each count in the sample by 3.2 to get estimated total in drawer.

Let’s do that:

- White: 12 × 3.2 = 38.4
- Blue: 1 × 3.2 = 3.2
- Black: 3 × 3.2 = 9.6
- Tan: 4 × 3.2 = 12.8

But again — we can’t have partial socks. However, since this is a prediction based on sampling, we might round to nearest whole number. But let’s check if adding them gives 64:

38.4 + 3.2 + 9.6 + 12.8 = 64 → yes, perfect.

But since socks must be whole numbers, perhaps we should report as decimals for accuracy in prediction, or round reasonably.

Actually, in real life, you’d probably round to nearest whole number. Let’s see:

White: 38.4 → 38
Blue: 3.2 → 3
Black: 9.6 → 10
Tan: 12.8 → 13

Check sum: 38 + 3 + 10 + 13 = 64 → perfect!

Alternatively, maybe the problem expects us to use fractions without rounding? But since it says “how many socks”, it implies whole numbers.

I think rounding to nearest whole number is acceptable here.

✔ So predicted socks in drawer:

White = 38
Blue = 3
Black = 10
Tan = 13

*(Note: Some teachers might prefer keeping decimal for intermediate steps, but final answer should be whole socks.)*

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Part C: How many pairs of each sock are in your drawer?

Since 1 pair = 2 socks, divide each sock count by 2.

From Part B:

- White: 38 ÷ 2 = 19 pairs
- Blue: 3 ÷ 2 = 1.5 pairs → wait, that’s not possible! You can’t have half a pair.

Uh oh — problem. If we rounded socks to whole numbers, we might end up with odd numbers, which don’t make pairs.

Let’s go back.

Maybe instead of rounding the sock counts first, we should calculate pairs directly from the probability.

Total pairs = 32.

Use same ratios:

Sample had 20 socks → corresponds to 32 pairs? No — 20 socks is 10 pairs worth? Wait no.

Actually, the 20 socks you pulled are individual socks, not pairs. And the total drawer has 64 individual socks = 32 pairs.

So to predict number of *pairs*, we can either:

Option 1: Predict individual socks first (as above), then divide by 2 → but may give halves.

Option 2: Use probability to predict proportion of pairs.

But note: the sample is of individual socks, so the probability reflects individual sock colors, not pairs.

However, since each pair consists of two matching socks, the proportion of colors among individual socks should match the proportion among pairs — assuming all socks are paired correctly (which they aren’t necessarily, but for prediction purposes, we assume the distribution is consistent).

So, for example, if 12 out of 20 socks are white, then about 12/20 of all socks are white → so 12/20 of 64 socks = 38.4 white socks → 19.2 pairs.

Again, we get decimals.

Perhaps the problem allows decimal predictions? Or maybe we should not round until the end.

Looking at Part D, it says actual values are given in pairs: 16 white pairs, etc. So likely, we are to predict pairs directly.

Alternative approach:

The 20 socks you drew represent a sample of the 64 socks.

So the fraction of white socks in sample = 12/20 = 3/5

So predicted white socks in drawer = (3/5)*64 = 38.4 → so predicted white pairs = 38.4 / 2 = 19.2

Similarly:

Blue: (1/20)*64 = 3.2 socks → 1.6 pairs
Black: (3/20)*64 = 9.6 socks → 4.8 pairs
Tan: (4/20)*64 = 12.8 socks → 6.4 pairs

Now, if we round these to nearest whole number for pairs:

White: 19.2 → 19 pairs
Blue: 1.6 → 2 pairs
Black: 4.8 → 5 pairs
Tan: 6.4 → 6 pairs

Check total pairs: 19 + 2 + 5 + 6 = 32 → perfect!

And this makes sense because pairs must be whole numbers.

In Part B, when we calculated socks, we had:

White: 38.4 → if we say 38 socks, that’s 19 pairs
Blue: 3.2 → if we say 3 socks, that’s 1.5 pairs — not good. But if we say 4 socks? That would be 2 pairs. But 4 socks would be 4/64 = 1/16, while sample was 1/20 — not matching.

Better to base pairs on the pair calculation.

Actually, let's think differently.

The key is: the sample of 20 socks gives us the proportion of each color. Since the total number of socks is 64, we can find expected number of socks per color, then convert to pairs.

But since pairs require even numbers, and our prediction might not be even, we have to accept that the prediction is approximate.

However, looking ahead to Part D, the actual values are:

- 16 pairs white → 32 socks
- 2 pairs blue → 4 socks
- 6 pairs black → 12 socks
- 8 pairs tan → 16 socks

Wait, 32+4+12+16=64 socks, good.

But in our sample, we had:

White: 12 out of 20 → 60%
Actual white socks: 32 out of 64 = 50% — not super close.

But anyway, for prediction, we should use the sample proportions.

I think the intended method is:

For Part B: multiply sample count by (64/20) = 3.2

So:

White: 12 * 3.2 = 38.4 → but since it's "how many socks", and socks are whole, perhaps report as 38 or 38.4? I think for prediction, decimal is ok, but typically in such problems, they expect you to round to nearest whole number.

But then for pairs, dividing by 2 might give half.

To avoid confusion, let's do this:

In Part B, calculate predicted socks as:

White: (12/20)*64 = 38.4 ≈ 38 socks
Blue: (1/20)*64 = 3.2 ≈ 3 socks
Black: (3/20)*64 = 9.6 ≈ 10 socks
Tan: (4/20)*64 = 12.8 ≈ 13 socks

Sum: 38+3+10+13=64 — good.

Then for Part C, pairs:

White: 38 / 2 = 19 pairs
Blue: 3 / 2 = 1.5 pairs — problem.

This is inconsistent.

Perhaps the problem expects us to keep the fractional part for accuracy, or maybe I made a mistake.

Another idea: maybe "predict the number" means to use the ratio to find how many of each color, and since it's a prediction, decimals are allowed.

But in reality, you can't have half a sock.

Let me check online or standard approach.

Upon second thought, in many textbooks, for such problems, they allow the predicted number to be decimal, as it's an estimate.

For example, if you flip a coin 10 times and get 6 heads, you predict 60% heads, so for 100 flips, 60 heads — even though it might not be exact.

So here, for Part B, we can leave as decimals or round.

But to match Part D, where actual are given as whole pairs, perhaps for Part C, we should predict pairs directly.

Let's try this:

The sample of 20 socks represents the composition.

Total pairs in drawer: 32.

But the sample is of individual socks, not pairs.

However, the proportion of colors among individual socks should be the same as among pairs, because each pair contributes two socks of the same color.

So, for example, if there are W white pairs, then there are 2W white socks.

Similarly for others.

So the proportion of white socks is 2W / 64 = W/32.

In the sample, proportion of white socks is 12/20 = 3/5.

So set W/32 = 3/5? No, that's not right.

The proportion in the sample should estimate the proportion in the population.

So proportion of white socks in population = 12/20 = 3/5.

But proportion of white socks = (number of white socks) / 64 = 2 * (number of white pairs) / 64 = (white pairs) / 32.

So:

(white pairs) / 32 = 12/20

Therefore, white pairs = 32 * (12/20) = 32 * 3/5 = 96/5 = 19.2

Similarly:

Blue pairs = 32 * (1/20) = 32/20 = 1.6

Black pairs = 32 * (3/20) = 96/20 = 4.8

Tan pairs = 32 * (4/20) = 128/20 = 6.4

So predicted pairs:

White: 19.2
Blue: 1.6
Black: 4.8
Tan: 6.4

Now, for reporting, since the question asks "how many pairs", and pairs are discrete, we might round to nearest whole number.

19.2 → 19
1.6 → 2
4.8 → 5
6.4 → 6

Sum: 19+2+5+6=32 — perfect.

And this matches what I had earlier.

For Part B, if we want socks, we can double these:

White socks: 19.2 * 2 = 38.4 → or from rounded pairs: 19*2=38
Blue: 1.6*2=3.2 → or 2*2=4? Inconsistency.

If we use the rounded pairs for Part C, then for Part B, socks would be twice that:

White: 19 pairs * 2 = 38 socks
Blue: 2 pairs * 2 = 4 socks
Black: 5 pairs * 2 = 10 socks
Tan: 6 pairs * 2 = 12 socks

Sum: 38+4+10+12=64 — good.

But in the sample, blue was only 1 out of 20, which is 5%, while 4 out of 64 is 6.25% — close.

Whereas if we used 3 socks for blue, that's 4.6875%, closer to 5%.

But 3 socks would be 1.5 pairs, which is not integer.

I think the best approach is to calculate the predicted number of pairs first using the formula:

Predicted pairs of color X = (sample count of X / total sample) * total pairs

So:

White pairs = (12/20) * 32 = (3/5)*32 = 96/5 = 19.2
Blue pairs = (1/20)*32 = 32/20 = 1.6
Black pairs = (3/20)*32 = 96/20 = 4.8
Tan pairs = (4/20)*32 = 128/20 = 6.4

Then for Part B, socks = 2 * pairs:

White socks = 2*19.2 = 38.4
Blue socks = 2*1.6 = 3.2
Black socks = 2*4.8 = 9.6
Tan socks = 2*6.4 = 12.8

Or, if we round the pairs to whole numbers first, then socks are even.

Given that in Part D, actual are given as whole pairs, and the prediction is to be compared, I think it's fine to report the predicted pairs as decimals for accuracy, or round them.

But let's see what the problem likely expects.

In many similar problems, they expect you to use the ratio to find the expected number, and report as is, or round.

To resolve, let's look at the numbers.

Sample: 12 white out of 20.

Total socks: 64.

Expected white socks = (12/20)*64 = 38.4

Since you can't have 0.4 sock, but for prediction, it's ok to say approximately 38 or 38.4.

But in the context, for Part B, "how many socks", they might want whole number, so round to nearest.

38.4 -> 38
3.2 -> 3
9.6 -> 10
12.8 -> 13

Sum 64.

Then for Part C, pairs: 38/2=19, 3/2=1.5, 10/2=5, 13/2=6.5 — not good.

3/2=1.5 pairs is invalid.

So perhaps for Part C, we should use the pair calculation directly and round the pairs.

I think the most reasonable way is:

For Part B: report predicted socks as:

White: 38.4 or 38
But to be precise, let's keep it as 38.4 for now, but since the answer blank is for a number, probably expect integer.

Perhaps the problem has a typo or expects us to use the sample to estimate, and accept decimals.

Another idea: maybe "predict the number" means to use the proportion to find how many, and since 20 socks were sampled, and 64 total, the multiplier is 3.2, so:

White: 12 * 3.2 = 38.4 — but perhaps write as 38 or 38.4.

I recall that in some curricula, they teach to round to nearest whole number for such predictions.

Moreover, in Part D, the actual are given, so we can compare.

Let's proceed with rounding the sock counts to nearest whole number for Part B, and for Part C, since pairs must be integer, we'll round the pair predictions to nearest whole number.

So for Part B:

White: 38.4 -> 38 socks
Blue: 3.2 -> 3 socks
Black: 9.6 -> 10 socks
Tan: 12.8 -> 13 socks

For Part C, pairs:

From socks: White 38/2 = 19 pairs
Blue 3/2 = 1.5 -> not good, so instead, from the pair calculation: 1.6 -> 2 pairs
Black 4.8 -> 5 pairs
Tan 6.4 -> 6 pairs

But then socks would be 4, 10, 12 for blue, black, tan, but we have 3,10,13 — inconsistency.

To avoid this, let's define:

For Part B: predicted number of socks = (sample count / 20) * 64, and round to nearest integer.

So:

White: (12/20)*64 = 38.4 -> 38
Blue: (1/20)*64 = 3.2 -> 3
Black: (3/20)*64 = 9.6 -> 10
Tan: (4/20)*64 = 12.8 -> 13

For Part C: predicted number of pairs = predicted socks / 2, but since socks may be odd, we can't.

So perhaps for Part C, calculate as (sample count / 20) * 32, and round to nearest integer.

So:

White pairs = (12/20)*32 = 19.2 -> 19
Blue pairs = (1/20)*32 = 1.6 -> 2
Black pairs = (3/20)*32 = 4.8 -> 5
Tan pairs = (4/20)*32 = 6.4 -> 6

Then for Part B, if we want socks, it would be 2* pairs: 38, 4, 10, 12 — sum 64, but then blue socks are 4, while in sample it was 1, and 4/64=6.25%, while 1/20=5%, close.

Whereas if we use 3 socks for blue, it's 4.6875%, also close.

But 3 socks means 1.5 pairs, which is not possible, so for consistency, perhaps the problem intends for us to use the pair prediction for Part C, and for Part B, use the sock prediction from the same logic.

I think I found a better way.

In Part B, "how many socks of each color are in your drawer?" — this is asking for the predicted number based on the experiment.

The experiment gave us the proportion.

So predicted number of white socks = (12/20) * 64 = 38.4

Since it's a prediction, and socks are discrete, but in statistics, we often report the expected value as decimal.

However, for school level, they might expect rounding.

Let's look at the numbers in Part D: actual are 32 white socks (16 pairs), 4 blue socks (2 pairs), 12 black socks (6 pairs), 16 tan socks (8 pairs).

Our sample had 12 white, 1 blue, 3 black, 4 tan.

So for white, sample 12/20=60%, actual 32/64=50% — overestimate.

Blue: sample 5%, actual 4/64=6.25% — underestimate.

etc.

For prediction, we should use the sample proportion.

I think for this problem, the expected answer is to calculate the predicted number as (sample count / 20) * 64 for socks, and for pairs, (sample count / 20) * 32, and report as decimals or round.

But to match the format, and since the blanks are for numbers, likely they want integers.

Perhaps calculate and round.

Let's do this:

For Part B:
White: 38.4 -> 38
Blue: 3.2 -> 3
Black: 9.6 -> 10
Tan: 12.8 -> 13

For Part C: since pairs, and 38 socks white -> 19 pairs
3 socks blue -> 1.5 pairs — not good, so perhaps for blue, since 3.2 socks, and 3.2/2 = 1.6 pairs, round to 2 pairs.

Similarly, black 9.6/2 = 4.8 -> 5 pairs
Tan 12.8/2 = 6.4 -> 6 pairs

Then for white, 38/2 = 19 pairs.

So Part C: White 19, Blue 2, Black 5, Tan 6

Sum 32.

And for Part B, if we use these pairs, socks would be 38, 4, 10, 12 — but we have 38,3,10,13 — so for blue and tan, discrepancy.

To resolve, perhaps in Part B, after calculating predicted socks, for Part C, divide by 2 and round to nearest integer, accepting that it may not sum to 32, but in this case it does if we round properly.

With socks: 38,3,10,13

Pairs: 19, 1.5, 5, 6.5 — sum 32, but 1.5 and 6.5 not integer.

Round 1.5 to 2, 6.5 to 7, then sum 19+2+5+7=33 — too many.

Round 1.5 to 1, 6.5 to 6, sum 19+1+5+6=31 — too few.

So not good.

Therefore, the best way is to calculate the predicted pairs directly as (sample count / 20) * 32, and round to nearest integer, and for socks, use 2* that.

So for Part B: socks = 2 * [(sample count / 20) * 32] = (sample count / 20) * 64 — same as before.

But when we round the pairs, the socks may not match the direct calculation.

I think for the sake of this problem, and since it's a worksheet, they likely expect:

Part A: probabilities as fractions or decimals.

Part B: predicted socks = (count/20)*64, and report as integer by rounding.

Part C: predicted pairs = predicted socks / 2, and if not integer, round or something.

But to make it work, let's use the following:

From sample, the ratio is 12:1:3:4 for white:blue:black:tan.

Total parts = 12+1+3+4=20.

Total socks = 64.

So each "part" corresponds to 64/20 = 3.2 socks.

So:

White: 12 * 3.2 = 38.4 -> 38 socks
Blue: 1 * 3.2 = 3.2 -> 3 socks
Black: 3 * 3.2 = 9.6 -> 10 socks
Tan: 4 * 3.2 = 12.8 -> 13 socks

For pairs, since 1 pair = 2 socks, but the number of pairs for each color is not simply socks/2 if socks are odd, but in reality, for prediction, we can say the number of pairs is approximately socks/2.

For white: 38/2 = 19 pairs
Blue: 3/2 = 1.5 -> perhaps 1 or 2, but let's say 1.5 is not allowed, so maybe the problem has a mistake, or we should use the pair calculation.

I recall that in some sources, for such problems, they calculate the expected number of pairs as (number of socks of color / 2) , but since it's prediction, use the proportion.

Let's calculate the predicted number of pairs as:

For a given color, the number of pairs is half the number of socks, but since the number of socks is predicted, and may be odd, it's problematic.

Perhaps the "pairs" in the drawer are not necessarily matched, but the question is "how many pairs of each color", implying how many complete pairs of that color.

In that case, if you have 3 blue socks, you can make 1 pair, with 1 left over.

So for prediction, if we predict 3 blue socks, then number of blue pairs = floor(3/2) = 1, but that seems complicated.

I think for this level, they expect us to assume that the number of socks is even, or to use the proportion for pairs directly.

Let's look at Part D: actual are given as 16,2,6,8 pairs.

Our sample suggests white is overrepresented.

For prediction, let's use:

Predicted pairs = (sample count / 20) * 32

So:

White: (12/20)*32 = 19.2
Blue: (1/20)*32 = 1.6
Black: (3/20)*32 = 4.8
Tan: (4/20)*32 = 6.4

Then for reporting, since the blank is for a number, and to compare with actual, we can leave as decimals or round.

In many worksheets, they round to nearest whole number.

So:

White: 19.2 -> 19
Blue: 1.6 -> 2
Black: 4.8 -> 5
Tan: 6.4 -> 6

Sum 32.

For Part B, socks = 2 * pairs = 38, 4, 10, 12

Sum 64.

And in the sample, blue was 1, which is 5%, while 4/64=6.25%, close; tan 4/20=20%, 12/64=18.75%, close.

Whereas if we used 3 socks for blue, it's 4.6875%, also close, but then pairs would be 1.5, which is not integer.

So I think for consistency, we should use the pair prediction for Part C, and for Part B, use 2* that.

So let's go with that.

So summary:

Part A:
P(white) = 12/20 = 3/5 or 0.6
P(blue) = 1/20 = 0.05
P(black) = 3/20 = 0.15
P(tan) = 4/20 = 1/5 or 0.2

Part B: predicted socks:
White: 2 * 19.2 = 38.4 -> but if we use rounded pairs, 2*19 = 38
Better to calculate from the beginning.

Define:

Let P_color = sample count / 20

Then predicted socks = P_color * 64

Predicted pairs = P_color * 32

Then for reporting, round predicted pairs to nearest integer for Part C, and for Part B, use 2 * rounded pairs, or use the direct calculation.

To minimize error, use direct calculation for socks, and for pairs, use direct calculation and round.

But for the answer, I'll do:

Part B:
White: (12/20)*64 = 38.4 -> 38
Blue: (1/20)*64 = 3.2 -> 3
Black: (3/20)*64 = 9.6 -> 10
Tan: (4/20)*64 = 12.8 -> 13

Part C:
White: 38/2 = 19
Blue: 3/2 = 1.5 -> since can't have half, and 3.2 socks suggests 1.6 pairs, round to 2
But to be consistent, perhaps calculate pairs as (sample count / 20) * 32 and round.

I think I'll box the answers as per common practice.

After research in my mind, I recall that in such problems, they often expect:

For Part B: multiply the sample count by 3.2 and round to nearest integer.

For Part C: divide the Part B result by 2 and round to nearest integer, but ensure sum is 32.

With socks: 38,3,10,13

Pairs: 19, 1.5, 5, 6.5

Round 1.5 to 2, 6.5 to 6 or 7? 6.5 usually rounds to 7, but 19+2+5+7=33>32.

Round 1.5 to 1, 6.5 to 6, sum 19+1+5+6=31<32.

So not good.

Therefore, the only way to have sum 32 is to use the pair calculation and round the pairs.

So:

Predicted pairs:
White: 19.2 -> 19
Blue: 1.6 -> 2
Black: 4.8 -> 5
Tan: 6.4 -> 6
Sum 32.

Then for Part B, socks = 2* pairs = 38, 4, 10, 12

Sum 64.

And in the sample, for blue, we had 1, which is low, but prediction is 4, which is higher, but based on proportion.

So I think this is acceptable.

For Part B, if we use 4 blue socks, while sample had 1, but 1/20=5%, 4/64=6.25%, while actual is 4/64=6.25%, so good.

In sample, tan was 4/20=20%, prediction 12/64=18.75%, actual 16/64=25%, so not great, but ok.

So I'll go with that.

So Final decision:

Part A:
P(white) = 12/20 = 3/5
P(blue) = 1/20
P(black) = 3/20
P(tan) = 4/20 = 1/5

Part B: predicted socks:
White: (12/20)*64 = 38.4 -> but since we'll use 2* pairs, and pairs are 19, so 38
Actually, to be accurate, let's calculate as:

From pairs prediction:

White pairs: 19.2 -> 19, so socks 38
Blue pairs: 1.6 -> 2, so socks 4
Black pairs: 4.8 -> 5, so socks 10
Tan pairs: 6.4 -> 6, so socks 12

So Part B: White = 38, Blue = 4, Black = 10, Tan = 12

Part C: White = 19, Blue = 2, Black = 5, Tan = 6

Part D: actual are 16 white pairs, 2 blue, 6 black, 8 tan.

So comparison:

White: predicted 19, actual 16 — over by 3
Blue: predicted 2, actual 2 — exact
Black: predicted 5, actual 6 — under by 1
Tan: predicted 6, actual 8 — under by 2

So accuracy: for white, off by 3; blue perfect; black off by 1; tan off by 2.

Overall, not very accurate for white and tan.

But let's write the answers.

So for the blanks:

Part A:
P(white) = \frac{12}{20} = \frac{3}{5} or 0.6
But probably simplify.

In the blank, likely fraction or decimal.

I think fraction is fine.

So:

P(white) = \frac{3}{5}
P(blue) = \frac{1}{20}
P(black) = \frac{3}{20}
P(tan) = \frac{1}{5}

Part B:
White = 38
Blue = 4
Black = 10
Tan = 12

Part C:
White = 19
Blue = 2
Black = 5
Tan = 6

Part D: Compare to actual: 16 white pairs, 2 blue, 6 black, 8 tan.

So prediction was:
White: 19 vs 16 — difference of 3
Blue: 2 vs 2 — difference 0
Black: 5 vs 6 — difference 1
Tan: 6 vs 8 — difference 2

So the prediction was fairly accurate for blue, less for others. Specifically, it overestimated white by 3 pairs, underestimated black by 1, tan by 2.

Now for the final answer, since the user wants the final answer after reasoning, and for the blanks, we need to fill in.

So let's box the answers as per the sections.

First, Part A:

P(white) = \frac{3}{5}
P(blue) = \frac{1}{20}
P(black) = \frac{3}{20}
P(tan) = \frac{1}{5}

Part B:
White = 38
Blue = 4
Black = 10
Tan = 12

Part C:
White = 19
Blue = 2
Black = 5
Tan = 6

Part D: The prediction was somewhat accurate for blue socks (exact match), but overestimated white socks by 3 pairs, underestimated black socks by 1 pair, and underestimated tan socks by 2 pairs. Overall, it was not very accurate for white and tan.

But for the blank in D, it says "How accurate was your prediction?" so probably a sentence.

In the worksheet, it's a blank, so perhaps write a short answer.

Since the user said "solve the problem accurately", and for D, it's descriptive.

For the final answer, perhaps list all.

But the instruction is to provide the final correct answer after reasoning.

So for the response, I'll write the answers for each part.

So Final Answer:

A. P(white) = \frac{3}{5}, P(blue) = \frac{1}{20}, P(black) = \frac{3}{20}, P(tan) = \frac{1}{5}
B. White = 38, Blue = 4, Black = 10, Tan = 12
C. White = 19, Blue = 2, Black = 5, Tan = 6
D. The prediction was accurate for blue socks (2 pairs predicted, 2 actual), but overestimated white socks (19 vs 16), underestimated black socks (5 vs 6), and underestimated tan socks (6 vs 8). The largest errors were for white and tan socks.

To make it neat.

For D, perhaps calculate the error.

But I think that's fine.

So I'll go with that.

Note: In Part B, I have Blue = 4, but in the sample it was 1, and 4 is from 2*2, while if we did direct, it was 3.2->3, but I chose 4 to match the pair prediction.

To confirm, let's see the actual: blue socks are 4 (2 pairs), so prediction of 4 is good.

In sample, we got only 1, which is luck, but prediction should be based on proportion.

So 1/20 of 64 = 3.2, which is closer to 3 than 4, but 3.2 is 3.2, and 4 is 4, difference 0.8, while 3 is difference 0.2, so 3 is closer.

3.2 - 3 = 0.2, 4 - 3.2 = 0.8, so 3 is closer.

But then for pairs, 3 socks mean 1.5 pairs, which is not integer.

Perhaps the problem allows for the prediction to be decimal, but for the blank, they want integer.

I think for school level, they might expect:

Part B: White: 38, Blue: 3, Black: 10, Tan: 13

Part C: White: 19, Blue: 1.5 or 2, but since can't, perhaps 1 or 2.

I found a solution online for similar problem, but since I can't, I'll decide.

Let's calculate the predicted number of pairs as the number that makes sense.

Another way: the number of pairs of a color is approximately (number of socks of that color in sample / 2) * (64/20) / 2 wait.

I think I'll stick with the initial direct calculation for socks, and for pairs, use socks/2 and round, and accept that sum may not be 32, but in this case with rounding, it can be.

With socks: 38,3,10,13

Pairs: 19, 1.5, 5, 6.5

If we round 1.5 to 2, 6.5 to 6, sum 19+2+5+6=32 — oh! 6.5 rounded to 6? Usually 6.5 rounds to 7, but in some contexts, to even, but 6 is even, 7 odd, but typically 6.5 rounds to 7.

But if we round 6.5 to 6, then sum is 32.

19+2+5+6=32.

And 1.5 to 2, 6.5 to 6.

Is that valid? 6.5 is exactly halfway, and sometimes rounded to nearest even, but 6 is even, 7 odd, so to 6.

In many schools, they teach to round 0.5 up, so 6.5 to 7.

But to make sum 32, perhaps force it.

With 1.5 to 1, 6.5 to 7, sum 19+1+5+7=32.

Also good.

So two ways:

Option 1: Blue pairs 2, Tan pairs 6, sum 32

Option 2: Blue pairs 1, Tan pairs 7, sum 32

Which is better?

Sample blue: 1 out of 20 = 5%
Actual blue: 4 out of 64 = 6.25%
Prediction: if 2 pairs = 4 socks = 6.25% — good
If 1 pair = 2 socks = 3.125% — worse.

Sample tan: 4/20=20%
Actual: 16/64=25%
Prediction: 6 pairs = 12 socks = 18.75% — close to 20%
7 pairs = 14 socks = 21.875% — also close.

But 12.8 socks predicted, so 6.4 pairs, so 6 or 7.

6.4 is closer to 6 than to 7? 6.4 - 6 = 0.4, 7 - 6.4 = 0.6, so closer to 6.

Similarly, blue 1.6, closer to 2 than to 1? 1.6-1=0.6, 2-1.6=0.4, so closer to 2.

So for blue, 1.6 -> 2
For tan, 6.4 -> 6

Sum 19+2+5+6=32.

Perfect.

And for socks, if we want, 38,4,10,12, but in Part B, if we put Blue=4, while direct calculation is 3.2, but 4 is from 2*2, and 2 is rounded from 1.6.

Direct calculation for blue socks is 3.2, which is closer to 3 than to 4, but for consistency with pairs, we use 4.

To avoid confusion, for Part B, use the direct calculation: 38,3,10,13

For Part C, use 19,2,5,6 (rounding the pair values)

Then for blue, socks 3, pairs 2 — but 2 pairs require 4 socks, contradiction.

So impossible.

Therefore, the only consistent way is to have the number of socks even for each color, or accept that the prediction for socks may not match the pair prediction.

For the purpose of this problem, I think the intended answer is:

Part B: White: 38, Blue: 3, Black: 10, Tan: 13

Part C: White: 19, Blue: 1, Black: 5, Tan: 6 (since 3/2=1.5->1, 13/2=6.5->6 or 7, but 6.5->6 if round down, but usually up)

With Blue: 1 pair (2 socks), but we have 3 socks, so not matching.

I think I have to choose one.

Let's look at the actual values in D: white 32 socks, blue 4, black 12, tan 16.

Our sample: white 12, blue 1, black 3, tan 4.

So for white, sample 12/20=60%, actual 32/64=50%, so prediction should be 50% of 64 = 32, but we have 38.4, so over.

For blue, sample 5%, actual 6.25%, prediction 3.2 or 4.

Perhaps the problem expects us to use the proportion without rounding for the calculation, but for the answer, put the number.

I recall that in some worksheets, they have students calculate the expected number as decimal, and leave it.

For example, P(white) = 12/20 = 0.6, then predicted white socks = 0.6 * 64 = 38.4, and they write 38.4.

Similarly for pairs, 0.6 * 32 = 19.2.

Then for D, compare 19.2 to 16, etc.

And for the blank, write 38.4, etc.

That might be it.

In that case, for Part B: White = 38.4, Blue = 3.2, Black = 9.6, Tan = 12.8

Part C: White = 19.2, Blue = 1.6, Black = 4.8, Tan = 6.4

Then for D, compare to actual 16,2,6,8.

So prediction was 19.2 vs 16 for white, etc.

And accuracy can be discussed.

And for the blank, they can write decimals.

In the worksheet, the blanks are lines, so likely allow decimals.

So I think that's the way.

So let's do that.

Part A: probabilities as fractions or decimals. I'll use fractions simplified.

P(white) = 12/20 = 3/5
P(blue) = 1/20
P(black) = 3/20
P(tan) = 4/20 = 1/5

Part B: predicted socks = (probability) * 64

So:
White = (3/5) * 64 = 192/5 = 38.4
Blue = (1/20) * 64 = 64/20 = 3.2
Black = (3/20) * 64 = 192/20 = 9.6
Tan = (1/5) * 64 = 64/5 = 12.8

Part C: predicted pairs = (probability) * 32

White = (3/5) * 32 = 96/5 = 19.2
Blue = (1/20) * 32 = 32/20 = 1.6
Black = (3/20) * 32 = 96/20 = 4.8
Tan = (1/5) * 32 = 32/5 = 6.4

Part D: actual pairs: white 16, blue 2, black 6, tan 8.

So prediction:
White: 19.2 vs 16 — error of 3.2
Blue: 1.6 vs 2 — error of 0.4
Black: 4.8 vs 6 — error of 1.2
Tan: 6.4 vs 8 — error of 1.6

So the prediction was closest for blue socks (error 0.4), least accurate for white (error 3.2).

Overall, not very accurate, especially for white and tan.

For the final answer, we can write the numbers.

So I think this is the most accurate way.

So Final Answer:

A. P(white) = \frac{3}{5}, P(blue) = \frac{1}{20}, P(black) = \frac{3}{20}, P(tan) = \frac{1}{5}
B. White = 38.4, Blue = 3.2, Black = 9.6, Tan = 12.8
C. White = 19.2, Blue = 1.6, Black = 4.8, Tan = 6.4
D. The prediction was most accurate for blue socks (predicted 1.6 pairs, actual 2 pairs, error 0.4), and least accurate for white socks (predicted 19.2 pairs, actual 16 pairs, error 3.2). For black and tan, errors were 1.2 and 1.6 respectively. Overall, the prediction had significant errors, especially for white and tan socks.

To box it.

Since the user may expect integers, but I think decimals are correct for prediction.

So I'll go with that.

For the response, I'll write the answers.

So here is the final answer.

Final Answer:

A. P(white) = \frac{3}{5}, P(blue) = \frac{1}{20}, P(black) = \frac{3}{20}, P(tan) = \frac{1}{5}
B. White = 38.4, Blue = 3.2, Black = 9.6, Tan = 12.8
C. White = 19.2, Blue = 1.6, Black = 4.8, Tan = 6.4
D. The prediction was fairly accurate for blue socks (close to actual 2 pairs), but overestimated white socks (19.2 vs 16), underestimated black socks (4.8 vs 6), and underestimated tan socks (6.4 vs 8). The largest error was for white socks.

Note: In some contexts, they might round, but for accuracy in prediction, decimals are appropriate.

To match the format, perhaps write the numbers as is.

I think this is correct.

So for the boxed answer, since it's multiple parts, I'll list them.

The user said "provide only the final correct answer", but for multi-part, probably all.

So I'll write:

\boxed{\begin{array}{c} \text{A. } P(\text{white}) = \dfrac{3}{5}, P(\text{blue}) = \dfrac{1}{20}, P(\text{black}) = \dfrac{3}{20}, P(\text{tan}) = \dfrac{1}{5} \\ \text{B. White} = 38.4, \text{ Blue} = 3.2, \text{ Black} = 9.6, \text{ Tan} = 12.8 \\ \text{C. White} = 19.2, \text{ Blue} = 1.6, \text{ Black} = 4.8, \text{ Tan} = 6.4 \\ \text{D. Prediction was most accurate for blue socks, least for white socks.} \end{array}}

For D, a bit vague, but ok.

To be precise.

Since the problem might expect integers, but I think decimals are fine.

I recall that in the actual values, they are integers, but prediction can be decimal.

So I'll submit that.

Final Answer

\boxed{\begin{array}{l} \text{A. } P(\text{white}) = \dfrac{3}{5}, P(\text{blue}) = \dfrac{1}{20}, P(\text{black}) = \dfrac{3}{20}, P(\text{tan}) = \dfrac{1}{5} \\ \text{B. White} = 38.4, \text{ Blue} = 3.2, \text{ Black} = 9.6, \text{ Tan} = 12.8 \\ \text{C. White} = 19.2, \text{ Blue} = 1.6, \text{ Black} = 4.8, \text{ Tan} = 6.4 \\ \text{D. The prediction was closest for blue socks (error 0.4), farthest for white socks (error 3.2).} \end{array}}

To make D better: "The prediction had errors: white +3.2, blue -0.4, black -1.2, tan -1.6 pairs compared to actual."

But perhaps for the box, keep it simple.

I think it's fine.

So this is the answer.