Answer:
Explanation:
In this scenario, we have two variables of interest: Y1 and Y2. Y1 represents the number of heads obtained after tossing three balanced coins independently, and Y2 represents the amount of money won on a side bet.
To calculate the expected value of Y1, we need to consider all possible outcomes of the three coin tosses. There are 2^3 = 8 possible outcomes, each of which has a probability of 1/8. The possible values of Y1 are 0, 1, 2, and 3. The expected value of Y1 is therefore:
E(Y1) = (01/8) + (13/8) + (23/8) + (31/8) = 1.5
This means that on average, we can expect to get 1.5 heads when tossing three balanced coins independently.
To calculate the expected value of Y2, we need to consider the probabilities of each possible outcome of the coin tosses and the corresponding payout for each outcome. We can break this down into four cases:
Case 1: The first head appears on the first toss. This has a probability of 1/2 (since the first toss can be either heads or tails). The payout for this outcome is $1.
Case 2: The first head appears on the second toss. This has a probability of (1/2)*(1/2) = 1/4 (since the first toss must be tails and the second toss must be heads). The payout for this outcome is $2.
Case 3: The first head appears on the third toss. This has a probability of (1/2)(1/2)(1/2) = 1/8 (since the first two tosses must be tails and the third toss must be heads). The payout for this outcome is $3.
Case 4: No heads appear. This has a probability of (1/2)(1/2)(1/2) = 1/8 (since each toss must be tails). The payout for this outcome is -$1.
The expected value of Y2 is therefore:
E(Y2) = (11/2) + (21/4) + (31/8) + (-11/8) = 7/8
This means that on average, we can expect to win $0.875 on the side bet when tossing three balanced coins independently.