Choose all the situations in which events A and B are independent.
P(A) = 0.5, P(B) = 0.4, P(A&B) = 0.9
P(A) = 0.3, P(B) = 0.6, P(A&B) = 0.18
P(A) = 0.1, P(B) = 0.1, P(A&B) = 0.01
P(A) = 0.3, P(B) = 0.3, P(A&B) = 0.2
does P(A&B) = P(A) * P(B)?
To determine if two events, A and B, are independent, we need to compare the product of their individual probabilities, P(A) and P(B), with the probability of their intersection, P(A&B).
If P(A) * P(B) = P(A&B), then A and B are independent.
Let's calculate for each situation:
1. P(A) = 0.5, P(B) = 0.4, P(A&B) = 0.9
P(A) * P(B) = 0.5 * 0.4 = 0.2 ≠ 0.9
A and B are not independent.
2. P(A) = 0.3, P(B) = 0.6, P(A&B) = 0.18
P(A) * P(B) = 0.3 * 0.6 = 0.18 = 0.18
A and B are independent.
3. P(A) = 0.1, P(B) = 0.1, P(A&B) = 0.01
P(A) * P(B) = 0.1 * 0.1 = 0.01 = 0.01
A and B are independent.
4. P(A) = 0.3, P(B) = 0.3, P(A&B) = 0.2
P(A) * P(B) = 0.3 * 0.3 = 0.09 ≠ 0.2
A and B are not independent.
Situations 2 and 3 have events A and B that are independent.