I have a question related to the wording of a basic probability problem which I have partially completed. Specifically, I have trouble wrapping my head around the English statement highlighted in bold in order to make a comparison.
The problem statement is as follows:
A coin is tossed twice. Alice claims that the event of two heads is at least as likely if we know that the first toss is a head than if we know that at least one of the tosses is a head. Is she correct?
My solution to the problem thus far is as follows:
Visualizing sample space of the random experiment makes it easy to compute the conditional probabilities.
$$ \Omega = \{(H1,H2),(H1,T2),(T1,H2),(T1,T2)\} $$
The probability of a second head given the first head happened is as follows:
$$ P(H2|H1) = \frac{P(two \; heads)}{P(head \; on \; the \; first \; toss)} =\frac{1/4}{2/4} = \frac{1}{2} $$
while the event another head happens given a head occus in the two coin toss happens is
$$ P(two \; heads | a \; head \; occurred ) = \frac{P(two \; heads \; occurring)}{P(a \; head \; occurred) } = \frac{1/4}{3/4} = \frac{1}{3} $$
Would it be possible for somebody to restate the problem statement for me in a different way? I do not understand how to compare the two likelihoods I found.
Best Answer
Alice claims that $P(H2|H1) \ge P(\text{two heads}|\text{a head occurred})$. You verified that.