There are events X, Y and Z. P(X) = 0.25, P(Y) = 0.04, and P(Z) = 0.3
P(X and Z) = 0.20, P(X and Z) = 0.075, and the P(Y and Z) = 0.015. Which of these two events are independent?
To determine if two events are independent, we need to compare the probability of their intersection to the product of their individual probabilities.
Let's compare events X and Z:
P(X and Z) = 0.20, P(X) = 0.25, P(Z) = 0.3
P(X and Z) = P(X) * P(Z)
0.20 = 0.25 * 0.3
Since the equation is true, events X and Z are independent.
Now let's compare events X and Y:
P(X and Y) = 0.075, P(X) = 0.25, P(Y) = 0.04
P(X and Y) = P(X) * P(Y)
0.075 ≠ 0.25 * 0.04
Since the equation is not true, events X and Y are not independent.
Therefore, event X and Z are independent, but event X and Y are not independent.
To determine if two events are independent, we need to check if the probability of their intersection is equal to the product of their individual probabilities.
Let's calculate the product of the individual probabilities for events X and Z:
P(X) = 0.25
P(Z) = 0.3
P(X) * P(Z) = 0.25 * 0.3 = 0.075
Now let's compare this to the probability of the intersection of events X and Z (given as P(X and Z) = 0.20).
Since P(X and Z) = 0.20 is not equal to P(X) * P(Z) = 0.075, events X and Z are not independent.
Now let's do the same calculation for events Y and Z:
P(Y) = 0.04
P(Z) = 0.3
P(Y) * P(Z) = 0.04 * 0.3 = 0.012
Next, let's compare this to the probability of the intersection of events Y and Z (given as P(Y and Z) = 0.015).
Since P(Y and Z) = 0.015 is approximately equal to P(Y) * P(Z) = 0.012, events Y and Z are independent.
Therefore, the events Y and Z are independent.