Question

Which of the following statements is false?

		If I have two mutually exclusive events A and B, then P(A|B)=0.
		Two independent events cannot occur at the same time.
		If I have two independent events A and B, then P(A|B)=P(A).
		Event A is independent of event B when the outcome of A has no impact on B.

Answer 1

Answer: Option 2 is false

Explanation:

1st option: Mutually exclusive events are the events which cannot occur at the same time.

For example,  the two possible outcomes of a coin flip are mutually exclusive, when you flip a coin, it cannot land both heads and tails simultaneously.

Let Event(A) = occurence of Head

Event(B) = occurence of Tail

Therefore, P(A|B) = Probability of A given that B has already occurred

Since B ( occurence of tail) has already occured, then event A (occurence of Head) is not possible as both head and tail cannot happen in a toss.

Therefore, P(A|B) = 0 (TRUE)

2nd Option:

Two independent Events can happen at the same time.

For example, Rolling of a die and tossing of a coin are two independent events which can happen at the same time. ( FALSE)

3rd Option:

If two events are independent, then

Probability of both the events A and B occuring = P(A and B) = P(A) *P(B)

Now, Conditional Probability:

P(A|B) = P(A and B) / P(B) (formula for conditional probability)

implies, P(A|B) = { P(A) *P(B) } / P(B)

Therefore, P(A|B) = P(A)

In other words, Probability of A given B has no effect on it because A is independent of B. (TRUE)

4th Option:

Just an extension of Option 3 result.

Since A is independent of B, it has no effect on the probability of A as prooved above. (TRUE)

ANSWER: Two independent events cannot occur at the same time is FALSE