Three coins are flipped. If at least one of them comes up heads, then what is the probability that they all come up heads?
The answer to this is 1/7, as it is the #(ways to have all heads)/#(ways to have at least one head).
If the question had asked, probability of all heads given that the first flip is heads, then the answer would have been 1/4 (because 1/2 * 1/2).
Why are these two answers different? Why is it that the "at least one" is a weaker condition, when at least simply means that one of the three coins is heads, which is the same as choosing a specific head, like the first coin? I would greatly appreciate some intuition here. Thanks!
Why is the at least one case less likely? If anything, shouldn't it be more likely because it doesn't restrict a specific coin to come up heads?