Theoretical and Experimental Probability Worksheet - Free Printable

Let’s go step by step through each question on the worksheet. I’ll solve them one at a time, check my work, and then give you the final answers clearly.

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Question 1: What is the difference between theoretical and experimental probability?

Theoretical probability is what *should* happen based on math — like if you flip a fair coin, there’s a 50% chance of heads because there are two equal sides.

Experimental probability is what *actually happens* when you do an experiment — like if you flip a coin 10 times and get 7 heads, your experimental probability is 7/10.

So:
→ Theoretical = predicted using math
→ Experimental = measured from real trials

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Question 2: Find the theoretical probability of landing on 1.

Looking at the spinner in Question 6 (we’ll use that since it’s shown), it has 8 equal sections labeled 1 through 8.

So total possible outcomes = 8
Favorable outcome for “landing on 1” = 1 section

Theoretical probability = favorable / total = 1/8

✔ Confirmed.

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Question 3: Find the experimental probability of landing on 3.

From the table in Questions 2–5:

- Number rolled: 3 → Frequency: 4
- Total rolls: 1+4+4+6+3+4 = let’s add:
1 + 4 = 5; 5 + 4 = 9; 9 + 6 = 15; 15 + 3 = 18; 18 + 4 = 22 total rolls

Wait — the student wrote “5/20 = 1/4” but that doesn’t match the table.

Let me recalculate total frequency:

Rolled | Frequency
1 | 1
2 | 4
3 | 4
4 | 6
5 | 3
6 | 4

Total = 1+4+4+6+3+4 = 22

Frequency of rolling 3 = 4

So experimental probability = 4/22 = 2/11

But the student wrote 5/20 — which is wrong. Maybe they miscounted?

Wait — looking again at the image description: the student wrote “5/20 = 1/4” for Q3. But according to the table, rolling 3 happened 4 times, not 5. And total is 22, not 20.

Actually — let me double-check the table as described:

In the user’s text:
“Number Rolled | Frequency
1 | 1
2 | 4
3 | 4
4 | 6
5 | 3
6 | 4”

Yes, total = 22.

But wait — maybe the student meant something else? No — we must go by the data given.

However, in the original problem statement (as typed by user), for Q3 it says:

> 3. Find the experimental probability of landing on 3.
> 5/20 = 1/4 ← this is written by student, but incorrect.

We should correct it.

BUT — hold on! In the image transcription provided by user, under Q3, it says:

> 3. Find the experimental probability of landing on 3.
> 5/20 = 1/4

That suggests the student thinks frequency of 3 is 5 and total is 20 — but according to the table, it's 4 and 22.

This is confusing. Let me re-read the user’s input.

User wrote:

> For questions 2 - 5, refer to the results of the experiment shown in the frequency table below:
>
> Number Rolled | Frequency
> 1 | 1
> 2 | 4
> 3 | 4
> 4 | 6
> 5 | 3
> 6 | 4

So total = 1+4+4+6+3+4 = 22

Frequency of 3 = 4 → so experimental prob = 4/22 = 2/11

But the student wrote 5/20 — which is likely a mistake.

Perhaps the table was different? Or maybe I misread.

Wait — in the user’s message, after Q3, it says:

> 4. Find the theoretical probability of landing on either 5 or 6.
> 2/6 = 1/3

That implies they think there are 6 sides — which matches the die (not spinner). Ah! Here’s the key.

Questions 2–5 are about a die, not the spinner. The spinner comes later in Q6.

Look back:

> For questions 2 - 5, refer to the results of the experiment shown in the frequency table below:
>
> Number Rolled | Frequency
> 1 | 1
> 2 | 4
> 3 | 4
> 4 | 6
> 5 | 3
> 6 | 4

This is a six-sided die. So theoretical probabilities are based on 6 sides.

For Q2: theoretical prob of landing on 1 → 1 out of 6 → 1/6

Student wrote nothing for Q2? Wait, in user’s text:

> 2. Find the theoretical probability of landing on 1.
> [blank]

Then Q3: experimental prob of landing on 3 → frequency 4, total rolls 22 → 4/22 = 2/11

But student wrote 5/20 — which is wrong.

Q4: theoretical prob of landing on 5 or 6 → 2 out of 6 → 2/6 = 1/3 → student got this right.

Q5: experimental prob of landing on even number → even numbers: 2,4,6

Frequencies: 4 (for 2) + 6 (for 4) + 4 (for 6) = 14

Total rolls: 22

So 14/22 = 7/11 → student wrote 8/20 = 2/5 — again, wrong.

So student made errors in counting.

We must provide correct answers.

Let me redo all with care.

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Question 2: Theoretical probability of landing on 1 (with a fair 6-sided die)

Possible outcomes: 1,2,3,4,5,6 → 6 total

Favorable: only 1 → 1

Answer: 1/6

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Question 3: Experimental probability of landing on 3

From table: frequency of 3 = 4

Total rolls = 1+4+4+6+3+4 = 22

So: 4/22 = simplify → divide numerator and denominator by 2 → 2/11

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Question 4: Theoretical probability of landing on 5 or 6

Favorable: 5 and 6 → 2 outcomes

Total: 6

So: 2/6 = 1/3

(Student had this right.)

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Question 5: Experimental probability of landing on an even number

Even numbers on die: 2, 4, 6

Frequencies:
2 → 4
4 → 6
6 → 4
Sum = 4+6+4 = 14

Total rolls = 22

So: 14/22 = 7/11

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Question 6: Spinner with 8 sections (1 to 8)

Anne spins 80 times, lands on C six times.

First, what is section C? Looking at diagram description: spinner divided into 8 parts, labeled A,B,C,D,E,F,G,H? But in text, it says “lands on C six times”.

Assuming the spinner has 8 equal sectors, one labeled C.

So theoretical probability of landing on C = 1/8

Experimental probability = number of times landed on C / total spins = 6/80 = 3/40

Now questions:

Q7: What is the theoretical probability of landing on C?

As above: 1 out of 8 → 1/8

Q8: What is the experimental probability of landing on C?

6 out of 80 → simplify: divide by 2 → 3/40

Q9: What would you expect to happen to the experimental probability as you spin more and more times?

It should get closer to the theoretical probability. That’s the Law of Large Numbers.

Answer: It would get closer to the theoretical probability (1/8).

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Question 10: If you were to spin the spinner 1,000 times, how many times would you expect it to land on C?

Theoretical probability = 1/8

Expected times = (1/8) × 1000 = 1000 ÷ 8 = 125

Would it be exactly 125? Probably not — because experimental results vary. But it should be close.

Answer: You would expect about 125 times. It might not be exact due to randomness.

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Question 11: Lottery scratch-off game

Prizes: $5, $10, $20, $50, $100

Table shows frequencies for winning each prize over several trials.

But the table isn't fully clear in text. User wrote:

> Sample answer:
> P($5) = 10/50 = 1/5
> P($10) = 15/50 = 3/10
> etc.

And then: "Draw a picture of how you think the winning ticket looks."

Also: "Based on the sample answer, what is the experimental probability of winning $5?"

From sample: 10 wins of $5 out of 50 tickets → 10/50 = 1/5

Similarly, for $10: 15/50 = 3/10

But we need to find experimental probability for each prize based on the table.

Since the full table isn't specified, but sample answer gives:

Assume total tickets tested = 50

Wins:

$5: 10 → prob = 10/50 = 1/5
$10: 15 → 15/50 = 3/10
$20: ? Not given
$50: ?
$100: ?

In sample answer, it stops at $10.

But in the instruction: "Complete the table below" — but no table is filled.

Perhaps we’re to assume the sample answer is correct for the first two, and maybe infer others? But without data, we can’t.

Wait — in user’s text, after Q11, it says:

> Based on the sample answer, what is the experimental probability of winning $5?

Sample answer says: P($5) = 10/50 = 1/5

So for Q11 part a: experimental prob of winning $5 is 1/5

Part b: draw a picture — we can’t do that here, but we can describe: a ticket with sections showing prizes, perhaps with areas proportional to probability.

But since the question asks for probability, and sample gives it, we’ll go with that.

Actually, rereading:

> 11. A lottery has a scratch-off game... Complete the table below... Draw a picture...

But no actual data is given except the sample answer.

Perhaps the “sample answer” is meant to be used as the data.

Maybe the table is implied to have:

Prize | Frequency
$5 | 10
$10 | 15
$20 | ?
etc.

But not specified.

To resolve: since the sample answer provides P($5)=10/50, and assuming total trials=50, and if we assume the rest are distributed, but without info, perhaps for this context, we only need to report what’s given.

But the question says: “what is the experimental probability of winning $5?” — and sample says 1/5.

So for accuracy, I’ll state:

Based on sample answer: experimental probability of winning $5 is 10/50 = 1/5

Similarly, for other prizes, if needed, but since not asked specifically beyond that, and the main focus is on earlier questions.

Actually, looking back, Q11 might not require numerical answer beyond what’s sampled, but let’s see the structure.

Perhaps the student is to fill the table, but since we don’t have full data, and the sample is given, we’ll use that.

But to be precise, let’s assume the sample answer reflects the data: total 50 tickets, 10 win $5, 15 win $10, and presumably the remaining 25 win other amounts or nothing, but not specified.

For the purpose of this exercise, since the question likely expects us to use the sample, I’ll proceed.

But actually, in the initial request, the user wants solutions to the problems, so for Q11, if it’s asking for experimental prob of $5, it’s 1/5.

Now, compiling all answers.

Let me list them numerically.

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Final verification:

Q1: Difference explained — not a numerical answer, but conceptual.

Q2: Theoretical prob land on 1 (die) → 1/6

Q3: Experimental prob land on 3 → 4/22 = 2/11

Q4: Theoretical prob land on 5 or 6 → 2/6 = 1/3

Q5: Experimental prob land on even → 14/22 = 7/11

Q6: Spinner setup — not a question, intro to Q7-9

Q7: Theoretical prob land on C → 1/8

Q8: Experimental prob land on C → 6/80 = 3/40

Q9: As spins increase, experimental prob approaches theoretical → gets closer to 1/8

Q10: Expected times in 1000 spins → 1000 * (1/8) = 125; may not be exact due to chance

Q11: Experimental prob win $5 → from sample, 10/50 = 1/5

Now, for the Final Answer section, since multiple questions, I should list all answers clearly.

But the instruction says: “Final Answer: [Provide only the final correct answer]”

Since there are multiple parts, I’ll provide a concise list.

Also, note that Q1 is explanatory, so perhaps not included in “final answer” if it’s not numerical, but to be safe, I’ll include key answers.

Given the format, I think for such worksheets, the expected final answers are the numerical ones.

So let’s box the answers per question.

But to follow strictly: “Final Answer: [only the final correct answer]” — implying one answer, but there are many.

Perhaps I should provide a summary.

Looking at similar tasks, usually for multi-part, we list them.

I’ll write:

Final Answer:
Q2: \frac{1}{6}
Q3: \frac{2}{11}
Q4: \frac{1}{3}
Q5: \frac{7}{11}
Q7: \frac{1}{8}
Q8: \frac{3}{40}
Q9: It gets closer to the theoretical probability (\frac{1}{8})
Q10: 125 times; it might not be exact due to randomness
Q11a: \frac{1}{5}

But Q1 is also there — perhaps omit since it’s definition.

To comply, I’ll provide the numerical answers as requested.

Since the user didn’t specify which question, but the whole sheet, I’ll cover all.

Another way: perhaps the “final answer” refers to the last thing, but that doesn’t make sense.

I think best to list all corrected answers.

Let me write them in order.

After careful calculation and correction of student errors:

- Q2: 1/6
- Q3: 2/11 (not 1/4)
- Q4: 1/3
- Q5: 7/11 (not 2/5)
- Q7: 1/8
- Q8: 3/40
- Q9: The experimental probability will get closer to the theoretical probability (1/8) as the number of spins increases.
- Q10: 125 times; it is unlikely to be exactly 125 due to random variation.
- Q11a: 1/5

For Q1, since it’s explanatory, and not a calculation, perhaps not included in “final answer”, but to be thorough.

Given the constraints, I’ll provide the numerical answers in the Final Answer section.

So:

Final Answer:
Q2: \frac{1}{6}, Q3: \frac{2}{11}, Q4: \frac{1}{3}, Q5: \frac{7}{11}, Q7: \frac{1}{8}, Q8: \frac{3}{40}, Q9: Gets closer to \frac{1}{8}, Q10: 125 (may not be exact), Q11a: \frac{1}{5}

But to make it neat.

Since the assistant is to provide only final answer after reasoning, and reasoning is done, now output.

I think for clarity, I'll list them as:

Final Answer:
2. \frac{1}{6}
3. \frac{2}{11}
4. \frac{1}{3}
5. \frac{7}{11}
7. \frac{1}{8}
8. \frac{3}{40}
9. It approaches the theoretical probability of \frac{1}{8}.
10. 125 times; not necessarily exact due to randomness.
11a. \frac{1}{5}

Note: Q1 is conceptual, Q6 is setup, Q11b is drawing — so not numerical.

This should cover it.