Final answer:
Jordan's expected gain from playing the basketball game at the carnival is -$1.00 per game, meaning he is expected to lose $1 on average each time he plays.
Step-by-step explanation:
Jordan is considering playing a basketball game where he has to make long-distance shots at a carnival. To model his potential earnings from the game, we need to calculate the expected value based on the given probabilities. Jordan has a 40% chance of making each shot and can take two shots per game. The cost per game is $5, earning back $5 for making one shot, and $10 for making both shots. Let's define a random variable X for the number of shots Jordan makes, where X can be 0, 1, or 2. Next, we will define M for the amount of money Jordan gains, which is dependent on X.
First, we calculate the probability of each outcome of X:
- P(X=0) = (1-0.4)2 = 0.36 (since Jordan misses both shots)
- P(X=1) = 2*(0.4)*(1-0.4) = 0.48 (since he can make either the first OR the second shot)
- P(X=2) = 0.42 = 0.16 (since Jordan makes both shots)
Next, we associate each value of X with the money gain M:
- M(X=0) = -$5 (since he loses the game cost)
- M(X=1) = $0 (since he gets his $5 back)
- M(X=2) = $5 (since he gains $5 after the $5 cost is returned)
To find the expected value (EV) of M, we multiply each value of M by the corresponding probability of X and sum the results:
E(M) = P(X=0)*M(X=0) + P(X=1)*M(X=1) + P(X=2)*M(X=2)
E(M) = 0.36*(-$5) + 0.48*($0) + 0.16*($5)
E(M) = -$1.80 + $0.00 + $0.80
E(M) = -$1.00
So, the expected monetary gain for Jordan is -$1.00 per game, which means, on average, Jordan will lose $1 every time he plays this basketball game at the carnival.