a) The marginal probability distribution for your winnings on the side bet is therefore:
Amount of money Probability
$3 3/8
$2 3/8
$1 1/8
-$1 1/8
b) The probability that you obtained three heads, given that you won $1 on the side bet, is 1.
a) Derive the marginal probability distribution for your winnings on the side bet.
The marginal probability distribution for your winnings on the side bet is the probability of winning each amount of money. To calculate this, we need to sum the probabilities of all the outcomes that result in that amount of money.
There are three ways to win $3: HHH, HHT, and HTH. The probability of each of these outcomes is 1/8. Therefore, the probability of winning $3 is 3(1/8) = 3/8.
There are three ways to win $2: TTH, THT, and HTT. The probability of each of these outcomes is 1/8. Therefore, the probability of winning $2 is 3(1/8) = 3/8.
There is one way to win $1: TTT. The probability of this outcome is 1/8. Therefore, the probability of winning $1 is 1/8.
There is one way to lose $1: TTH. The probability of this outcome is 1/8. Therefore, the probability of losing $1 is 1/8.
The marginal probability distribution for your winnings on the side bet is therefore:
Amount of money Probability
$3 3/8
$2 3/8
$1 1/8
-$1 1/8
b) What is the probability that you obtained three heads, given that you won $1 on the side bet?
The probability of obtaining three heads, given that you won $1 on the side bet, is the conditional probability of HHH given that you won $1. This can be calculated using the following formula:
P(HHH | win $1) = P(HHH and win $1) / P(win $1)
We know that P(win $1) = 1/8, since this is the probability of winning $1 on the side bet. We also know that P(HHH and win $1) = 1/8, since this is the probability of obtaining three heads and winning $1 on the side bet. Therefore, the probability of obtaining three heads, given that you won $1 on the side bet, is:
P(HHH | win $1) = (1/8) / (1/8) = 1
Therefore, the probability that you obtained three heads, given that you won $1 on the side bet, is 1.