Answer: Therefore, the equation P(A and B) = P(A) * P(B) justifies whether events are dependent or independent by calculating the probability of both events occurring based on the individual probabilities of each event.
Explanation:
The equation that justifies whether events are dependent or independent is called the multiplication rule. This rule states that if two events, A and B, are independent, then the probability of both events occurring is the product of their individual probabilities.
Mathematically, the multiplication rule can be expressed as:
P(A and B) = P(A) * P(B)
Here's an example to illustrate how the multiplication rule works:
Let's say we have a bag with 5 red marbles and 3 blue marbles. We want to calculate the probability of drawing a red marble and then drawing a blue marble without replacement.
First, we calculate the probability of drawing a red marble:
P(Red) = Number of red marbles / Total number of marbles = 5 / 8
Next, we calculate the probability of drawing a blue marble after drawing a red marble:
P(Blue|Red) = Number of blue marbles / Remaining number of marbles = 3 / 7
To find the overall probability of both events occurring, we apply the multiplication rule:
P(Red and Blue) = P(Red) * P(Blue|Red) = (5/8) * (3/7) = 15/56