You roll a die. If an odd number comes up, you lose. If you get a 6, you win $54. If it is an even number other than 6, you get to roll again. If you get a 6 the second time, you win $18. If not, you lose.
(a) Construct a probability model for the amount you win at this game. Explain briefly how you obtain the probabilities
associated with the different
amounts of winning.
Let X be the payoff.
E(X) = $54(1/6) + $18(1/2) + $0(1/2) +$0(1/2)
= $18
not sure if that's correct
Hmmm. I see it as
$54(1/6) + (1/2)($18(1/6)) = $12
are you even trying? #QUTISCOMINGFORYOU
To construct a probability model for the amount you can win at this game, we need to consider all possible outcomes and their associated probabilities.
1. If you roll an odd number (1, 3, or 5), you lose. The probability of rolling an odd number is 3/6 or 1/2. Therefore, the amount you win in this outcome is $0, with a probability of 1/2.
2. If you roll a 6 on the first roll, you win $54. The probability of rolling a 6 is 1/6. Therefore, the amount you win in this outcome is $54, with a probability of 1/6.
3. If you roll an even number other than 6 on the first roll, you get to roll again. Now, we need to consider the second roll's outcomes:
3a. If you roll a 6 on the second roll, you win $18. The probability of rolling a 6 on the second roll is also 1/6, as you have already rolled an even number other than 6. Therefore, the amount you win in this outcome is $18, with a probability of (1/6) * (1/2) = 1/12, as the probability of reaching this outcome is (1/2) multiplied by the probability of rolling a 6 on the second roll, which is 1/6.
3b. If you don't roll a 6 on the second roll, you lose. The probability of not rolling a 6 on the second roll is 5/6 (as there are 5 outcomes out of 6 that are not a 6 on a fair die). Therefore, the amount you win in this outcome is $0, with a probability of 5/6 * 1/2 = 5/12, as the probability of not rolling a 6 on the second roll is 5/6 multiplied by the probability of reaching this outcome, which is 1/2.
Now, to find the expected value (E(X)) of the amount you can win, we multiply each outcome's amount by its respective probability and sum them up:
E(X) = ($0 * 1/2) + ($54 * 1/6) + ($18 * 1/12) + ($0 * 5/12)
= $0 + $9 + $1.5 + $0
= $10.5
Therefore, the expected value, or average amount you can win in this game, is $10.5.