Given the following probabilities, for what P(B) are events A and B independent?
P(A)=1/2 P(A and B)=1/6
1/4
1/8
3/4
1/3
you need P(A) * P(B) = P(A&B)
so it's 1/3, correct?
Never mind! It's the correct answer, thank you for pointing me in the right direction! :)
To determine whether events A and B are independent, we need to verify if the probability of event A occurring is unaffected by the occurrence of event B.
According to the definition of independence, two events A and B are independent if and only if P(A and B) = P(A) * P(B).
Let's use this definition to determine the value of P(B) for which events A and B are independent.
Given:
P(A) = 1/2
P(A and B) = 1/6
Plug these values into the definition of independence:
1/6 = (1/2) * P(B)
Now, we can solve for P(B):
Multiply both sides of the equation by 2 to eliminate the fraction:
2 * (1/6) = 1/2 * P(B)
1/3 = 1/2 * P(B)
Cross-multiply to solve for P(B):
2 * (1/3) = 1 * P(B)
2/3 = P(B)
Therefore, events A and B are independent when P(B) = 2/3.