Question

In an experiment, the probability that event a occurs is 0.4 and the probability that event b occurs is 0.5. If a and b are independent events, which statement is true?

1) The probability that both events a and b occur is 0.2
2) The probability that either event a or event b occurs is 0.9
3) The probability that event a occurs given that event b has occurred is 0.4
4) The probability that event b occurs given that event a has occurred is 0.5

Answer 1

In an experiment with independent events A and B, the probability of both A and B occurring is 0.2, the probability of either A or B occurring is 0.7, and the probabilities of A given B and B given A remain unchanged by the occurrence of the other event.

Step-by-step explanation:

The question involves the concepts of independent events and probability in mathematics. Considering the given probabilities that events A and B will occur independently, the following statements can be evaluated:

• The probability that both events A and B occur is the product of their individual probabilities: P(A AND B) = P(A) × P(B) = 0.4 × 0.5 = 0.2. So the first statement is true.
• For independent events, the probability that either event A or event B occurs is found using the addition rule: P(A OR B) = P(A) + P(B) – P(A AND B) = 0.4 + 0.5 – 0.2 = 0.7. Therefore, the second statement is false.
• Because A and B are independent, the probability of event A occurring given that event B has occurred is just P(A), and the same goes for B given A. So the third and fourth statements are true.