Practice Conditional Probability in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The conditional probability P(A∣B)P(A|B) is the probability of event AA occurring given that event BB has already occurred.

If I know BB happened, what's the chance of AA? Updates probability with new info.

Showing a random 20 of 54 problems.

Example 1

medium
A family has 33 children. Given the eldest is a girl, P(all three are girls)=?P(\text{all three are girls})=? Assume independent and equally likely.

Example 2

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70%70\% of customers want coffee; among them, 30%30\% also want pastry. P(coffee and pastry)=?P(\text{coffee and pastry})=?

Example 3

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A two-way table: 4040 students total. 2424 took the bus, 1616 walked. Among bus riders, 1818 ate breakfast. P(breakfast∣bus)=?P(\text{breakfast}|\text{bus})=?

Example 4

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A box has 5 red and 3 blue. Two are drawn without replacement. P(2nd red∣1st red)=?P(\text{2nd red}|\text{1st red})=?

Example 5

easy
P(B)=0.5P(B) = 0.5 and P(A∣B)=0.6P(A|B) = 0.6. Find P(A∩B)P(A \cap B).

Example 6

challenge
Three urns equally likely. Urn contents: P(red)=0.2,0.5,0.8P(\text{red})=0.2,0.5,0.8. A red is drawn. P(urn 3∣red)=?P(\text{urn 3}|\text{red})=?

Example 7

medium
Show why P(A∣B)≠P(B∣A)P(A|B) \ne P(B|A) in general by example: P(A)=0.5,P(B)=0.1,P(A∩B)=0.05P(A)=0.5,P(B)=0.1,P(A\cap B)=0.05. Find both.

Example 8

challenge
A pair of dice is rolled until the sum is 7 or 11. Find P(the stopping sum is 11)P(\text{the stopping sum is 11}).

Example 9

easy
P(A∣B)=P(A∩B)P(B)P(A|B)=\frac{P(A\cap B)}{P(B)}. What must be true of P(B)P(B) for this to be defined?

Example 10

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Among 10001000 patients, 100100 have disease DD. A test has 90%90\% sensitivity and 80%80\% specificity. Of those testing positive, how many actually have DD?

Example 11

easy
A coin is flipped twice. Given the first flip is heads, what is P(second is heads)P(\text{second is heads})?

Example 12

medium
P(A)=0.6P(A)=0.6, P(B)=0.5P(B)=0.5, P(A∩B)=0.3P(A\cap B)=0.3. Are A,BA,B independent? Answer 11 for yes.

Example 13

easy
If AA and BB are independent and P(A)=0.4P(A) = 0.4, find P(A∣B)P(A|B).

Example 14

challenge
A randomly chosen integer from 1 to 1000 is divisible by 5. Find P(also divisible by 6∣divisible by 5)P(\text{also divisible by 6}\mid\text{divisible by 5}).

Example 15

medium
A bag has 55 red, 33 green, 22 yellow. Two drawn without replacement. P(second is red∣first is green)=?P(\text{second is red}|\text{first is green})=?

Example 16

medium
P(snow)=0.2P(\text{snow})=0.2, P(accident∣snow)=0.4P(\text{accident}|\text{snow})=0.4, P(accident∣no snow)=0.05P(\text{accident}|\text{no snow})=0.05. P(accident)=?P(\text{accident})=?

Example 17

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Two cards are drawn without replacement from a standard deck. Given that the first card is a king, what is the probability the second card is also a king?

Example 18

challenge
A family has 22 children. Given that one of them is a boy born on a Tuesday, P(both are boys)=?P(\text{both are boys})=? (Independent, equally likely days and sexes.)

Example 19

hard
Three coins flipped. Given at least two heads, P(all three heads)=?P(\text{all three heads})=?

Example 20

easy
P(A∩B)=0.2P(A\cap B)=0.2 and P(B)=0.5P(B)=0.5. Find P(A∣B)P(A|B).
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