If events A and B are independent, what must be true?
[tex]$P(A)=P(B)$[/tex]
[tex]$P(A \mid B )=P(B)$[/tex]
[tex]$P(A \mid B )=P(A)$[/tex]
[tex]$P(A \mid B )=P(B \mid A )$[/tex]
Answer (2)
Independent events A and B satisfy P ( A an d B ) = P ( A ) × P ( B ) .
Conditional probability is defined as P ( A ∣ B ) = P ( B ) P ( A an d B ) .
For independent events, P ( A ∣ B ) = P ( B ) P ( A ) × P ( B ) = P ( A ) .
Therefore, the correct condition is P ( A ∣ B ) = P ( A ) .
Explanation
Analyze the problem We are given that events A and B are independent, and we need to determine which of the given conditions must be true. Let's analyze the properties of independent events and conditional probability to find the correct answer.
Recall definitions of independence and conditional probability Two events A and B are independent if and only if the probability of both A and B occurring is the product of their individual probabilities: P ( A an d B ) = P ( A ) × P ( B ) Conditional probability is defined as the probability of event A occurring given that event B has already occurred: P ( A ∣ B ) = P ( B ) P ( A an d B ) , provided that 0"> P ( B ) > 0 .
Derive the condition for independent events If A and B are independent, we can substitute the independence condition into the conditional probability formula: P ( A ∣ B ) = P ( B ) P ( A ) × P ( B ) As long as 0"> P ( B ) > 0 , we can cancel out P ( B ) from the numerator and denominator: P ( A ∣ B ) = P ( A ) This shows that if A and B are independent, the probability of A occurring given that B has occurred is simply the probability of A occurring.
Check the given conditions Now let's examine the given conditions:
P ( A ) = P ( B ) : This is not necessarily true for independent events. For example, flipping a coin and rolling a die are independent events, but the probabilities of getting heads on the coin and rolling a 6 on the die are different.
P ( A ∣ B ) = P ( B ) : This is not necessarily true. We derived that P ( A ∣ B ) = P ( A ) for independent events, and P ( A ) is not necessarily equal to P ( B ) .
P ( A ∣ B ) = P ( A ) : This is true, as we derived above.
P ( A ∣ B ) = P ( B ∣ A ) : This is not necessarily true. If P ( A ∣ B ) = P ( A ) and P ( B ∣ A ) = P ( B ) , then this condition would imply P ( A ) = P ( B ) , which is not always the case for independent events.
State the final answer Therefore, the condition that must be true if events A and B are independent is: P ( A ∣ B ) = P ( A ) This means that knowing whether event B has occurred does not change the probability of event A occurring.
Examples
Consider a scenario where you flip a coin and roll a die. The outcome of the coin flip does not affect the outcome of the die roll, and vice versa. These are independent events. If event A is getting heads on the coin flip and event B is rolling a 6 on the die, then P(A|B) = P(A) because knowing that you rolled a 6 doesn't change the probability of getting heads on the coin.
When events A and B are independent, the correct condition is P ( A ∣ B ) = P ( A ) . This means that the occurrence of event B does not change the probability of event A occurring. Thus, if we know B happens, the probability of A remains the same as it was before knowing B happened.
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