Let’s go through each problem one by one, step by step.
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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’re told the
first die is already a 4. So we only care about the second die.
The second die must also be a 4 for both to be 4s.
A die has 6 sides: 1, 2, 3, 4, 5, 6 → so chance of rolling a 4 is
1 out of 6.
✔ Final answer for #1:
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?
We know the
first quarter is tails. We need the other two to also be tails.
Each coin flip is independent — meaning what happens with one doesn’t affect the others.
Chance of tails on one coin = 1/2
So for the next two coins:
(1/2) × (1/2) =
1/4
✔ Final answer for #2:
1/4
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Now let’s look at the Venn diagram questions (#3–8).
From the diagram:
- Circle A (only): 21
- Overlap (A and B): 3
- Circle B (only): 33
- Outside both: 12
Total students = 21 + 3 + 33 + 12 =
69
But for conditional probability, we focus on the “given” part.
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Problem 3: Find P(A|B)
This means: Probability of A
given B happened.
Formula: P(A|B) = (number in A AND B) / (total in B)
In B total = overlap + only B = 3 + 33 =
36
A and B together =
3
So P(A|B) = 3 / 36 =
1/12
✔ Final answer for #3:
1/12
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Problem 4: Find P(B|A)
Probability of B given A.
P(B|A) = (A and B) / (total in A)
Total in A = only A + overlap = 21 + 3 =
24
A and B =
3
So P(B|A) = 3 / 24 =
1/8
✔ Final answer for #4:
1/8
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Problem 5: If A = 9th graders, B = students with pets, what does P(A|B) represent?
P(A|B) = probability someone is a 9th grader
given that they have a pet.
So it’s asking: Among students who have pets, what fraction are 9th graders?
✔ Final answer for #5:
The probability that a student is in 9th grade, given that they have a pet.
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Problem 6: What does P(B|A) represent?
P(B|A) = probability someone has a pet
given that they are in 9th grade.
So among 9th graders, what fraction have pets?
✔ Final answer for #6:
The probability that a student has a pet, given that they are in 9th grade.
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Problem 7: What do the 12 students not in A or B represent?
They are NOT 9th graders AND do NOT have pets.
So they’re students from other grades who don’t have pets.
✔ Final answer for #7:
Students who are not in 9th grade and do not have pets.
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Problem 8: How many students have pets?
Pets = set B = only B + overlap = 33 + 3 =
36
✔ Final answer for #8:
36
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Final Answer:
1. 1/6
2. 1/4
3. 1/12
4. 1/8
5. The probability that a student is in 9th grade, given that they have a pet.
6. The probability that a student has a pet, given that they are in 9th grade.
7. Students who are not in 9th grade and do not have pets.
8. 36
Parent Tip: Review the logic above to help your child master the concept of high school probability worksheet.
