Hello I have a question regarding dependency.
If a fair coin is tossed 6 times, and there is two events: A-there are more heads than tails given that B-the 6th toss is a head.
We are trying to find the conditional probability. However, I am confused as to whether these events are independent or dependent. I believe they are dependent because if B is true, then there needs to be at least 3 heads in the first 5 tosses. In other words, the minimum heads in the first 5 depend on whether B is true or not.
Is this correct logic?
Good thought.
Event $A$ occurs if $B$ occurs and there at least three heads among the first five tosses, or if $B$ does not occur and there are at least four heads among the first five tosses.
Let $X$ be the count for heads among the first five tosses.
$$\begin{align}\mathsf P(A\mid B) = \mathsf P(X\geq 3)\\\mathsf P(A\mid B^\complement) = \mathsf P(X\geq 4)\\\therefore \mathsf P(A\mid B)-\mathsf P(A\mid B^\complement)=\mathsf P(X=3)\end{align}$$
Since the event of throwing exactly 3 heads among the first five tosses has a non-zero probability, therefore these conditional probabilities are not equal, and so...