Conditional Probability Practice Worksheet with Venn Diagram and Probability Questions

Let’s go step by step through each problem.

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Problem 1:
> What is the probability of two dice landing on 4 if you already know the first die has landed on 4?

We are told the first die is already 4. So we only care about the second die.

A standard die has 6 sides: 1, 2, 3, 4, 5, 6.

The chance that the second die lands on 4 is 1 out of 6.

So the answer is:
→ 1/6

*(Note: The question says “two dice landing on 4” — but since the first is already known to be 4, we’re really just asking for the second die to also be 4. That’s conditional probability: P(both 4 | first is 4) = P(second is 4) = 1/6.)*

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Problem 2:
> What is the probability of three quarters landing on tails if you know the first one has landed on tails?

Again, we’re given that the first quarter is tails. We now need the next two quarters to also land on tails.

Each coin flip is independent. Probability of tails on one quarter = 1/2.

So for the next two:
P(tails and tails) = (1/2) × (1/2) = 1/4

Answer:
→ 1/4

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Now, use the Venn diagram for problems 3–8.

From the Venn diagram:

- Only in A (not B): 21
- In both A and B: 3
- Only in B (not A): 33
- Outside both A and B: 12

Total students = 21 + 3 + 33 + 12 = 69

Set A total = 21 + 3 = 24
Set B total = 3 + 33 = 36

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Problem 3: Find P(A|B)

This means: *Probability of being in A, given that you are in B.*

Formula:
P(A|B) = Number in both A and B / Total in B
= 3 / 36
= 1/12

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Problem 4: Find P(B|A)

This means: *Probability of being in B, given that you are in A.*

Formula:
P(B|A) = Number in both A and B / Total in A
= 3 / 24
= 1/8

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Problem 5:
> IF A represents 9th graders and B represents students that have pets, what does P(A|B) represent in the context of this problem?

P(A|B) = Probability a student is a 9th grader given that they have a pet.

So it’s:
→ *Among all students who have pets, what fraction are 9th graders?*

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Problem 6:
> What does P(B|A) represent in the context of this problem?

P(B|A) = Probability a student has a pet given that they are a 9th grader.

So it’s:
→ *Among all 9th graders, what fraction have pets?*

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Problem 7:
> What do the 12 students who are not in sets A or B represent in the context of this problem?

They are not 9th graders AND do not have pets.

So:
→ *Students who are NOT 9th graders and do NOT have pets.*

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Problem 8:
> How many students have pets?

That’s everyone in set B: those only in B (33) plus those in both A and B (3).

33 + 3 = 36

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Final Answer:

1. 1/6
2. 1/4
3. 1/12
4. 1/8
5. Among students who have pets, the probability that a student is a 9th grader.
6. Among 9th graders, the probability that a student has a pet.
7. Students who are not 9th graders and do not have pets.
8. 36