Conditional probability worksheet¦KS4 maths¦Teachit - Free Printable

Let’s solve this step by step.

We are asked:
What is the probability that Mary will be wearing yellow today, given that she is not wearing her favourite earrings?

This is a conditional probability:
P(yellow | not earrings)

We’ll use Bayes’ Theorem or the definition of conditional probability:

> P(A | B) = P(A and B) / P(B)

So here:
- A = wearing yellow
- B = not wearing earrings

So we need to find:
→ P(yellow and not earrings)
→ P(not earrings)

Then divide them.

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Step 1: Define events and probabilities from the problem

Let’s define:

- S = sunny → P(S) = 0.45
- Not S = not sunny → P(Not S) = 1 - 0.45 = 0.55

- Y = wears yellow
- P(Y | S) = 0.7
- P(Y | Not S) = 0.2

- E = wears favourite earrings
- P(E | Y) = 0.25 → so P(not E | Y) = 0.75
- P(E | not Y) = 0.10 → so P(not E | not Y) = 0.90

We want:
P(Y | not E)

By formula:
P(Y | not E) = P(Y and not E) / P(not E)

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Step 2: Find P(Y and not E)

To get “Y and not E”, we consider both weather cases (sunny and not sunny), because whether it’s sunny affects whether she wears yellow.

So:

P(Y and not E) =
P(Y and not E | S) * P(S) + P(Y and not E | Not S) * P(Not S)

But note:
P(Y and not E | S) = P(not E | Y, S) * P(Y | S)
But since earrings depend only on whether she wears yellow (not directly on weather), we can say:

P(not E | Y) = 0.75 regardless of weather.

So:

P(Y and not E | S) = P(not E | Y) * P(Y | S) = 0.75 * 0.7 = 0.525
Then multiply by P(S): 0.525 * 0.45 = 0.23625

Similarly:

P(Y and not E | Not S) = P(not E | Y) * P(Y | Not S) = 0.75 * 0.2 = 0.15
Then multiply by P(Not S): 0.15 * 0.55 = 0.0825

So total P(Y and not E) = 0.23625 + 0.0825 = 0.31875

Wait — but in the worksheet, one box says:

P(A ∩ B) = P(yellow ∩ not earrings) = 0.23625 + _______

That suggests they’re breaking it into parts. Let’s check their structure.

Actually, looking at the tree diagram hint, they might be computing all four paths for “not earrings”.

Let’s compute P(not E) fully first.

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Step 3: Compute P(not E) — total probability of not wearing earrings

She doesn’t wear earrings in two main cases:
1. She wears yellow AND doesn’t wear earrings
2. She does NOT wear yellow AND doesn’t wear earrings

And each of those depends on weather.

So let’s compute all four combinations that lead to “not earrings”:

#### Case 1: Sunny, Yellow, Not Earrings
P(S) * P(Y|S) * P(not E|Y) = 0.45 * 0.7 * 0.75 =
0.45 * 0.525 = 0.23625

#### Case 2: Sunny, Not Yellow, Not Earrings
P(S) * P(not Y|S) * P(not E|not Y)
P(not Y|S) = 1 - 0.7 = 0.3
P(not E|not Y) = 0.90
So: 0.45 * 0.3 * 0.9 = 0.45 * 0.27 = 0.1215

#### Case 3: Not Sunny, Yellow, Not Earrings
P(Not S) * P(Y|Not S) * P(not E|Y) = 0.55 * 0.2 * 0.75 =
0.55 * 0.15 = 0.0825

#### Case 4: Not Sunny, Not Yellow, Not Earrings
P(Not S) * P(not Y|Not S) * P(not E|not Y)
P(not Y|Not S) = 1 - 0.2 = 0.8
P(not E|not Y) = 0.90
So: 0.55 * 0.8 * 0.9 = 0.55 * 0.72 = 0.396

Now, total P(not E) = sum of all four cases where she doesn’t wear earrings:

= 0.23625 + 0.1215 + 0.0825 + 0.396

Let’s add step by step:

0.23625 + 0.1215 = 0.35775
0.35775 + 0.0825 = 0.44025
0.44025 + 0.396 = 0.83625

So P(not E) = 0.83625

Earlier, P(Y and not E) = Case 1 + Case 3 = 0.23625 + 0.0825 = 0.31875

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Step 4: Compute P(Y | not E)

P(Y | not E) = P(Y and not E) / P(not E) = 0.31875 / 0.83625

Let’s calculate that.

First, write as fraction to avoid rounding error:

0.31875 ÷ 0.83625

Multiply numerator and denominator by 100000 to eliminate decimals:

31875 / 83625

Simplify:

Divide numerator and denominator by 75:

31875 ÷ 75 = 425
83625 ÷ 75 = 1115

Wait, better to divide by 25 first:

31875 ÷ 25 = 1275
83625 ÷ 25 = 3345

Again by 15:

1275 ÷ 15 = 85
3345 ÷ 15 = 223

So 85/223 ≈ ?

Let me do decimal division:

0.31875 ÷ 0.83625

Use calculator-style:

0.31875 / 0.83625 = ?

Compute:

≈ 0.3811...

Wait, let me do exact:

0.31875 / 0.83625 = ?

Note: 0.31875 = 31875/100000 = 51/160? Wait, better to use fractions from start.

Original values:

P(Y and not E) = 0.23625 + 0.0825 = 0.31875
= 31875/100000 = simplify:

Divide numerator and denominator by 125:
31875 ÷ 125 = 255
100000 ÷ 125 = 800
→ 255/800 = 51/160

P(not E) = 0.83625 = 83625/100000
Divide by 125: 669/800

Wait:

83625 ÷ 125 = 669? 125*669 = 125*(600+69)=75000+8625=83625 → yes.

So P(not E) = 669/800

P(Y and not E) = 51/160 = 255/800

So P(Y | not E) = (255/800) / (669/800) = 255/669

Simplify 255/669:

Divide numerator and denominator by 3:

255 ÷ 3 = 85
669 ÷ 3 = 223

So 85/223

Now compute decimal: 85 ÷ 223

223 × 0.38 = 223×0.3=66.9, 223×0.08=17.84 → total 84.74
85 - 84.74 = 0.26 → so ≈ 0.38116...

So approximately 0.381

But let’s check if the worksheet expects an exact fraction or decimal.

Looking back at the worksheet boxes:

One box says:

P(B) = P(not earrings) = 0.23625 + ___ + ___ + ___ = ___

Which matches our four cases: 0.23625 (sunny,yellow,notE) + 0.1215 (sunny,notY,notE) + 0.0825 (notS,yellow,notE) + 0.396 (notS,notY,notE) = 0.83625

Another box:

P(A ∩ B) = P(yellow ∩ not earrings) = 0.23625 + 0.0825 = 0.31875

Then final box:

P(yellow | not earrings) = 0.31875 / 0.83625 = ?

Let me compute numerically:

0.31875 ÷ 0.83625 = ?

Do the division:

0.83625 × 0.381 = ?

0.83625 × 0.3 = 0.250875
0.83625 × 0.08 = 0.0669
0.83625 × 0.001 = 0.00083625
So 0.3 + 0.08 + 0.001 = 0.381 → sum: 0.250875 + 0.0669 = 0.317775 + 0.00083625 ≈ 0.31861125 — very close to 0.31875

Difference: 0.31875 - 0.31861125 = 0.00013875

So try 0.3812:

0.83625 × 0.0002 = 0.00016725 → too big

Better: 0.31875 / 0.83625 = 31875 / 83625 = as before 85/223

85 ÷ 223 = ?

223 goes into 85 zero times.

850 ÷ 223 ≈ 3 (223×3=669) → remainder 181
1810 ÷ 223 ≈ 8 (223×8=1784) → rem 26
260 ÷ 223 ≈ 1 → rem 37
370 ÷ 223 ≈ 1 → rem 147
etc.

So 0.38116... → rounds to 0.381

But perhaps keep as fraction? The worksheet has blanks for numbers, likely decimal.

In the box: P(yellow | not earrings) = _____ / _____ = _____

So probably: 0.31875 / 0.83625 = ?

Let me compute exactly:

0.31875 ÷ 0.83625 = 31875 ÷ 83625

Divide numerator and denominator by 375:

31875 ÷ 375 = 85
83625 ÷ 375 = 223

Same as before.

So 85/223 is exact.

But for school level, maybe decimal is fine.

Check with calculator: 85 ÷ 223 = 0.381165919...

So to 3 decimal places: 0.381

But let’s confirm if we did everything right.

Alternative approach: Use the values given in the worksheet hints.

In the worksheet, one box says:

P(B) = P(not earrings) = 0.23625 + ___ + ___ + ___

We have:

0.23625 (from sunny, yellow, not earrings)
+ 0.1215 (sunny, not yellow, not earrings)
+ 0.0825 (not sunny, yellow, not earrings)
+ 0.396 (not sunny, not yellow, not earrings)
= 0.83625

Another box:

P(A ∩ B) = P(yellow ∩ not earrings) = 0.23625 + 0.0825 = 0.31875

Then P(yellow | not earrings) = 0.31875 / 0.83625

Now, let's compute this division precisely:

0.31875 ÷ 0.83625

Multiply numerator and denominator by 100000:

31875 / 83625

Reduce by dividing numerator and denominator by 75:

31875 ÷ 75 = 425
83625 ÷ 75 = 1115

Again by 5:

425 ÷ 5 = 85
1115 ÷ 5 = 223

So 85/223

As decimal: 85 ÷ 223

Let me do long division:

223 ) 85.000000

223 > 85, so 0.

850 ÷ 223 = 3 (since 223*3=669) → subtract: 850-669=181

Bring down 0 → 1810

1810 ÷ 223 = 8 (223*8=1784) → 1810-1784=26

Bring down 0 → 260

260 ÷ 223 = 1 → 260-223=37

Bring down 0 → 370

370 ÷ 223 = 1 → 370-223=147

Bring down 0 → 1470

1470 ÷ 223 = 6 (223*6=1338) → 1470-1338=132

So far: 0.38116...

So approximately 0.381

For most purposes, 0.381 is sufficient.

But let’s see what the worksheet might expect. In the box, it says:

P(yellow | not earrings) = _____ / _____ = _____

So likely: 0.31875 / 0.83625 = 0.381

Or perhaps they want the fraction.

But since other numbers are decimals, probably decimal.

I think 0.381 is acceptable.

But let me double-check the addition for P(not E):

Case 1: 0.45 * 0.7 * 0.75 = 0.45 * 0.525 = let's compute: 0.45 * 0.5 = 0.225, 0.45 * 0.025 = 0.01125, total 0.23625 ✓

Case 2: 0.45 * 0.3 * 0.9 = 0.45 * 0.27 = 0.1215 ✓

Case 3: 0.55 * 0.2 * 0.75 = 0.55 * 0.15 = 0.0825 ✓

Case 4: 0.55 * 0.8 * 0.9 = 0.55 * 0.72 = 0.396 ✓

Sum: 0.23625 + 0.1215 = 0.35775; +0.0825=0.44025; +0.396=0.83625 ✓

P(Y and not E) = case1 + case3 = 0.23625 + 0.0825 = 0.31875 ✓

Division: 0.31875 / 0.83625 = ?

Let me compute using fractions:

0.31875 = 31875/100000 = 51/160? Earlier we had 51/160 for 0.31875? 51÷160=0.31875 yes.

0.83625 = 83625/100000 = 669/800? 669÷800=0.83625 yes.

So (51/160) / (669/800) = (51/160) * (800/669) = (51 * 800) / (160 * 669)

Simplify: 800 / 160 = 5, so = (51 * 5) / 669 = 255 / 669

Then 255 ÷ 3 = 85, 669 ÷ 3 = 223 → 85/223

85/223 ≈ 0.381165919...

So to three decimal places, 0.381

I think that's the answer.

Final Answer:
0.381