Question

Use the axioms of probability to prove the probability of a complement of an event is 1 - the probability of the event.

Answer 1

The probability of the complement of an event A is equal to 1 minus the probability of event A. This can be proven using the axioms of probability.

Step-by-step explanation:

The probability of the complement of an event A, denoted as A', is equal to 1 minus the probability of A. This can be proven using the axioms of probability.

1. Let B be the complement event of A.
2. By definition, B consists of all outcomes that are not in A.
3. Since A and B together make up the entire sample space, the probability of A plus the probability of B is equal to 1.
4. Therefore, P(A) + P(B) = 1.
5. But B is the complement of A, so B = A'.
6. Substituting this into the equation, we get P(A) + P(A') = 1.
7. Simplifying further, we have P(A') = 1 - P(A).

Examples:
1. Let's say we have a fair coin. The event A is getting heads. The probability of getting heads is 0.5. The complement of A, A', is getting tails. The probability of getting tails is also 0.5. Therefore, 1 - P(A) = 1 - 0.5 = 0.5, which is the same as P(A').
2. Let's say we have a deck of cards. The event A is drawing a heart. The probability of drawing a heart is 1/4. The complement of A, A', is drawing a non-heart card. The probability of drawing a non-heart card is 3/4. Therefore, 1 - P(A) = 1 - 1/4 = 3/4, which is the same as P(A').