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Say we model tossing a coin with two outcomes: H,T such that $P(H) = P(T) = \frac{1}{2} $. If we now toss two such coins, the sample space would be $\Omega = \{ HH, HT, TH, TT \} $. Assuming tosses are independent,

What is the probability that both coins show a head given that the first shows a head ? what about the probability that both coins show heads given that at least one of them is a head??

TRY:

Let $A$ be the event that both coins show a head. $A = \{ HT, TH \} \subset \Omega$ and let $B $ be the event that first coin is a head. I want to find $P( A | B ) $. Since $A,B$ are independent, then $P(A | B) = P(A) = |A|/|\Omega| = 2/4 = 1/2 $.

Next, let $A$ be event that both coins show head so $A = \{ HH \} $ and $B$ be event that one of them is a head. so $P(A | B) = P(A) = 1/2 $

Is this correct?

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Let $A=\{HH\}$ be the event that both coins are head and $B=\{ HH,HT \}$ the event that the first shows head. Then, $$ P(A\mid B) = \frac{P(A \cap B)}{P(B)}=\frac{\frac{1}{4}}{\frac{2}{4}}= \frac{1}{2}.$$

For your second question, keep $A$ and change $B$ to $B=\{HH,HT,TH\}$. Then, $$ P(A\mid B) = \frac{P(A \cap B)}{P(B)}=\frac{\frac{1}{4}}{\frac{3}{4}}= \frac{1}{3}.$$

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