Question

a basketball team is down by 2 points with 10 seconds left. they can attempt a 2 - point shot with an 80% chance of success, or a 3 - point shot with a 40% chance of success. if they make the 2 - point shot, the game goes to overtime. if they make the 3 - point shot, they win. which option has a higher expected value in terms of winning the game? expected value = probability × shot value
2 - point shot
3 - point shot
they are equally likely to win.
it depends on the opponents skill.

Explanation:

Answer:

1. First, calculate the expected - value for the 2 - point shot:

• The probability of making the 2 - point shot is \(P_2 = 0.8\). If they make the 2 - point shot, the game goes to overtime. In overtime, assume the probability of winning is \(0.5\). So the probability of making the 2 - point shot and then winning in overtime is \(0.8\times0.5 = 0.4\). If they miss the 2 - point shot (with probability \(1 - 0.8=0.2\)), they lose. The expected value of the 2 - point shot for winning the game is \(E_2=0.8\times0.5+0.2\times0 = 0.4\).

1. Then, calculate the expected - value for the 3 - point shot:

• The probability of making the 3 - point shot is \(P_3 = 0.4\), and if they make it, they win. If they miss (with probability \(1 - 0.4 = 0.6\)), they lose. The expected value of the 3 - point shot for winning the game is \(E_3=0.4\times1+0.6\times0 = 0.4\).

So the answer is: They are equally likely to win.