Question 1
A coin is flipped three times. What is the probability of getting exactly two heads, given that at least one head occurred?
Explanation: This is a conditional probability problem, which asks for the probability of one event happening given that another event has already occurred. When you see "given that" in a probability question, you need to use the conditional probability formula or adjust your sample space.
First, let's find all possible outcomes when flipping a coin three times. There are total outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.
Since we're given that at least one head occurred, we can eliminate TTT from our sample space. This leaves us with 7 favorable outcomes for the condition: HHH, HHT, HTH, HTT, THH, THT, TTH.
Now we need exactly two heads from these 7 remaining outcomes. Looking at our list: HHT, HTH, and THH each contain exactly two heads. That's 3 outcomes with exactly two heads.
Therefore, the probability is .
Choice A () incorrectly uses the original sample space of 8 outcomes instead of the reduced space of 7. Choice C () correctly identifies 3 favorable outcomes but fails to account for the conditional restriction, using all 8 original outcomes as the denominator. Choice D () likely comes from incorrectly thinking there are only 6 remaining outcomes or miscounting the favorable cases.
Remember: in conditional probability problems, always adjust your sample space to include only the outcomes that satisfy the given condition, then find your target event within that restricted space.