To find the probability of event [tex]\( B \)[/tex] given that events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, we can use the definition of independent events. For two independent events [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ P(A \text{ and } B) = P(A) \times P(B) \][/tex]
We are given two pieces of information:
1. The probability of event [tex]\( A \)[/tex] occurring, [tex]\( P(A) = 0.64 \)[/tex]
2. The probability of both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occurring together, [tex]\( P(A \text{ and } B) = 0.24 \)[/tex]
We can rearrange the formula to solve for [tex]\( P(B) \)[/tex]:
[tex]\[ P(B) = \frac{P(A \text{ and } B)}{P(A)} \][/tex]
Substitute the given values into the equation:
[tex]\[ P(B) = \frac{0.24}{0.64} \][/tex]
Perform the division:
[tex]\[ P(B) = 0.375 \][/tex]
Therefore, the probability of event [tex]\( B \)[/tex] is:
[tex]\[ P(B) = 0.375 \][/tex]
Hence, to the nearest thousandth, the probability of event [tex]\( B \)[/tex] is [tex]\( 0.375 \)[/tex].
