Final answer
a) The probability of both events a and b occurring simultaneously (P(a ∩ b)) when a and b are independent is 0.1764 (rounded to 5 decimal places).
Step-by-step explanation:
When events are independent, the probability of both events happening together is found by multiplying their individual probabilities (P(a) * P(b)). For independent events a and b, the formula is P(a ∩ b) = P(a) * P(b).
Given:
P(a) = 0.36
P(b) = 0.49
Using the formula for independent events:
P(a ∩ b) = P(a) * P(b) = 0.36 * 0.49 = 0.1764
Therefore, the probability of both events a and b occurring simultaneously is 0.1764 when a and b are independent.
This calculation is derived from the fundamental principle of probability for independent events. When two events are independent, the occurrence of one event doesn't influence the probability of the other occurring. Hence, to find the probability of both events happening together, we multiply the probabilities of each event occurring individually.
In this case, with the given probabilities of events a and b, the multiplication of their probabilities provides the probability of both events occurring concurrently. The product, 0.1764, represents the likelihood of events a and b happening simultaneously when they are independent of each other.