Final answer:
On average, it takes 4 coin tosses for a winner to be determined in the game between Anna and Brenda.
Step-by-step explanation:
To find the expected number of coin tosses for a winner to be determined, we can consider the probability of each outcome at each toss. Let's start with the first toss: there are two possibilities, either a head or a tail. Since the coin is fair, the probability of getting a head or a tail is 1/2.
If Anna wins, we need to consider the probability of getting 2 heads in the first two tosses and then a 3rd head. The probability of getting 2 heads is (1/2)^2 = 1/4, and the probability of getting a 3rd head is 1/2. Multiplying these two probabilities, we get (1/4)*(1/2) = 1/8.
Similarly, if Brenda wins, we need to consider the probability of getting 2 tails in the first two tosses and then a 3rd tail. The probability of getting 2 tails is (1/2)^2 = 1/4, and the probability of getting a 3rd tail is 1/2. Multiplying these two probabilities, we get (1/4)*(1/2) = 1/8.
Since both Anna and Brenda winning are exclusive events, we can add their probabilities to find the total probability of winning on the first toss: 1/8 + 1/8 = 1/4. So, on average, it takes 4 tosses for a winner to be determined.