Theoretical Basic Probability Trivia Quiz

Answer: 0.7

Mutually exclusive events are events whose intersection is the empty set. So the probability of their union is just the sum of their probabilities, which is .7.

Answer: 0.7

The probability of an event and its complement must sum to 1, since they are mutually exclusive and their union is the entire sample space S. So the probability of the complement of E is 1 - .3 = .7.

Answer: 0.7

For any two sets, the probability of their union is just the sum of their probabilities, minus the probability of their intersection. This is called the principle of inclusion/exclusion. The reason you have to subtract the probability of the intersection is because the intersection of the two events is itself a subset of each event, so its probability gets counted twice. So the probability we're after is just .2 + .6 - .1 = .7.

Answer: 0.12

If E and F are independent events, the probability of their intersection is found by multiplying their respective probabilities. So the answer in this case is .3 * .4 = .12.

Answer: 0.58

By the principle of inclusion/exclusion, the probability of the union of E and F is the probability of E plus the probability of F minus the probability of the intersection of E and F. But E and F are independent, so the probability of their intersection is just .3 * .4 = .12. Hence the probability of their union is .3 + .4 - .12 = .58.

Answer: 0

Recall that mutually exclusive events are events whose intersection is the empty set. The empty set has probability 0.

Answer: 0.1

Note that E is the disjoint union of (E intersect (the complement of F)) and (E intersect F). Since the probability of E is .3 and the probability of (E intersect F) is .2, the probability we want is just .3 - .2 = .1.

Answer: 0.6

The union of E and (the complement of F) can be written as the disjoint union of E and ((the complement of E) intersect (the complement of F)). By the principle of inclusion/exclusion, the probability of the union of E and F is just the probability of E plus the probability of F minus the probability of the intersection of E and F. So the probability of the union of E and F is .5 + .6 - .2 = .9. Now by one of DeMorgan's laws, the complement of the union of E and F is the intersection of their complements, thus the probability of ((the complement of E) intersect (the complement of F)) is just 1 - .9 = .1. So the probability we want is (from the first sentence) .5 + .1 = .6. Note that this problem could also be solved by drawing a Venn diagram.

Answer: 0.8

The intersection of E and F is a subset of both E and F, hence its probability must be less than or equal to each of these probabilities. So the largest it can be is the smallest of the probabilities of E or F, which is .8.

Answer: 0.7

By the principle of inclusion/exclusion, the probability of the union of E and F is equal to the probability of E plus the probability of F minus the probability of E intersect F. Let x = the probability of E intersect F. Then E union F has probability .8 + .9 - x = 1.7 - x. But probabilities cannot be greater than 1: 1.7 - x less than or equal to 1 implies 1.7 - 1 less than or equal to x implies .7 less than or equal to x. Thus .7 is the smallest possible value for the probability of the intersection of E and F.

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