Thank you. I get it now. :Djohng said:Rearrange your equation to:
$$P(A)-P(A\cap B)=P(A\cup B)-P(B)$$
I hope you now see that
$$P(A\cap B)\leq P(A)$$
Now finish.
The equation represents the maximum value of the probability of two events, A and B, occurring together. It is equal to the minimum value between the individual probabilities of A and B.
This equation is useful in determining the likelihood of two events occurring simultaneously. It helps in calculating joint probabilities and understanding the relationship between two events.
Sure, let's say we are interested in the probability of both flipping a coin and rolling a dice and getting a heads and an even number, respectively. The equation would be: Max Value of $P(coin=H \cap dice=even): \min(P(coin=H),P(dice=even))
If the probabilities of A and B are equal, the equation simplifies to: Max Value of $P(A \cap B): P(A) = P(B)
No, this equation can be extended to more than two events. For example, for three events A, B, and C, the equation would be: Max Value of $P(A \cap B \cap C): \min(P(A),P(B),P(C))