Final answer:
To find the probability that Alice and Bob get an equal number of heads, we must account for all possible outcomes of their separate coin tosses and then sum the probabilities where they get the same number of heads.
Step-by-step explanation:
The probability that Alice and Bob get an equal number of heads when Alice throws a fair coin 4 times and Bob throws it 3 times can be calculated by considering all possible outcomes. Alice can get 0, 1, 2, 3, or 4 heads in her 4 coin tosses. For each of these possible outcomes for Alice, we calculate the probability that Bob also gets that number of heads in his 3 coin tosses.
For instance, the probability that Alice gets 2 heads is the number of ways to get 2 heads out of 4 coin tosses (which is 6 ways) divided by 16 (the total number of outcomes for 4 coin tosses). Similarly, we calculate the probability for Bob to get the same number of heads out of 3 coin tosses. Finally, we add the probabilities of these matching scenarios to get the total probability of Alice and Bob getting an equal number of heads.
Here's the calculation in detail:
- Alice gets 0 heads (1 way); Bob must also get 0 heads (1 way): P(A=0 and B=0) = (1/16) * (1/8).
- Alice gets 1 head (4 ways); Bob must get 1 head (3 ways): P(A=1 and B=1) = (4/16) * (3/8).
- Alice gets 2 heads (6 ways); Bob gets 2 heads (3 ways): P(A=2 and B=2) = (6/16) * (3/8).
- Alice gets 3 heads (4 ways); Bob gets 3 heads (1 way): P(A=3 and B=3) = (4/16) * (1/8).
- Alice gets 4 heads (1 way); Bob cannot get 4 heads as he only tosses 3 times: P(A=4 and B=4) = 0.
Adding up these probabilities gives us the total probability of Alice and Bob getting an equal number of heads.