Final Answer:
For independent events A and B, the probability of both occurring is found by multiplying their individual probabilities, i.e., P(A and B) = P(A) * P(B), resulting in 0.216 in this case.
Step-by-step explanation:
The probability of the intersection of two independent events, denoted as P(A and B), is calculated by multiplying the individual probabilities of each event. In this case, P(A and B) = P(A) * P(B). Given that events A and B are independent, the probability of both occurring is the product of their individual probabilities.
For event A, P(A) = 0.4, and for event B, P(B) = 0.54. To find the probability of both A and B happening, multiply these probabilities: 0.4 * 0.54 = 0.216. Therefore, the final answer is P(A and B) = 0.216.
This result makes intuitive sense for independent events because the occurrence of one event does not affect the occurrence of the other. The probability of both events happening is the product of their individual probabilities. In this scenario, approximately 21.6% of the time, both events A and B will occur simultaneously based on the given probabilities.