Practice Independent Events in Math

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

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Quick Recap

Two events are independent if the occurrence of one does not change the probability of the other: P(A∩B)=P(A)⋅P(B)P(A \cap B) = P(A) \cdot P(B).

They don't 'know about' each other. One happening tells you nothing about the other.

Showing a random 20 of 55 problems.

Example 1

medium
For two events AA and BB, P(A)=0.4P(A) = 0.4 and P(B)=0.5P(B) = 0.5. If AA and BB are independent, find (a) P(A∩B)P(A \cap B) and (b) P(A∪B)P(A \cup B).

Example 2

medium
Independent events A,BA,B: P(A)=0.6P(A)=0.6, P(A∩B)=0.24P(A\cap B)=0.24. Find P(B)P(B).

Example 3

hard
Events AA and BB satisfy P(A)=0.6P(A) = 0.6, P(B)=0.7P(B) = 0.7, and P(A∩B)=0.42P(A \cap B) = 0.42. Are AA and BB independent? Justify algebraically.

Example 4

challenge
Three independent components each fail with probability 0.10.1. P(at least one fails)?

Example 5

challenge
If AA and BB are independent, show AA and B′B' (complement) are also independent for P(A)=0.5P(A)=0.5, P(B)=0.4P(B)=0.4.

Example 6

easy
A bag has 33 red and 22 blue marbles. You draw one, replace it, then draw again. Find P(both red)P(\text{both red}).

Example 7

medium
A light bulb has probability 0.020.02 of failing in a year, independent of others. A fixture uses 33 bulbs. Find P(at least one fails)P(\text{at least one fails}).

Example 8

hard
A test for a virus has 98%98\% sensitivity. If three independent tests are run on a true positive, find P(all three tests positive)P(\text{all three tests positive}) and P(at least one test misses)P(\text{at least one test misses}).

Example 9

medium
A factory has two independent quality checks, each catching defects with probability 0.90.9. Find P(at least one check catches a given defect)P(\text{at least one check catches a given defect}).

Example 10

challenge
A,B,CA, B, C are pairwise independent, each with probability 1/21/2, but P(A∩B∩C)=1/4P(A\cap B\cap C) = 1/4, not 1/81/8. Are A,B,CA, B, C mutually independent?

Example 11

hard
A circuit has three independent switches, each closed with probability 0.90.9. Current flows only if ALL three are closed. Find P(current flows)P(\text{current flows}).

Example 12

medium
P(at least one head in 3 independent flips)?

Example 13

hard
Two independent shooters hit a target with probabilities 0.70.7 and 0.40.4. They each take one shot. Find P(exactly one hit)P(\text{exactly one hit}).

Example 14

medium
A free throw shooter makes 80% independently. P(makes first two, misses third)?

Example 15

medium
If P(A)=0.3P(A)=0.3 and P(B)=0.5P(B)=0.5 are independent, find P(A∪B)P(A \cup B).

Example 16

challenge
P(A)=pP(A)=p, P(B)=0.5P(B)=0.5, independent, and P(A∪B)=0.7P(A\cup B)=0.7. Find pp.

Example 17

easy
For independent AA and BB with P(A)=0.5P(A)=0.5, P(B)=0.8P(B)=0.8, find P(A∩B)P(A \cap B).

Example 18

easy
A student randomly guesses on two multiple-choice questions, each with 4 options. Find the probability of getting both correct.

Example 19

easy
A fair coin is flipped and a fair die is rolled. Find P(Heads and a 4)P(\text{Heads and a 4}).

Example 20

hard
If AA and BB are independent with P(A)=0.4P(A)=0.4, P(B)=0.7P(B)=0.7, find P(Ac∩Bc)P(A^c \cap B^c).
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