Now suppose Streak-Shooting Shelly has moved to mars. Shelly is still adjusting to Mars. Her overall performance is down, but she still shoots better when she has just made a shot.
Suppose Shelly has a 60% probability of making the first shot. If she gets the first shot, she has an 80% probability of making the second shot. If she gets the first two shots, her probability of making the third shot rises to 90%. How many pints is shelly most likely to scorn one-and-one-and-one situation?
How are "pints" awarded?
To determine how many points Shelly is most likely to score in a one-and-one-and-one situation, we need to consider the probabilities of her making each shot and the number of points she can score in each scenario.
Let's break it down step by step:
First scenario: Shelly makes the first shot.
- The probability of making the first shot is 60% (0.6).
- If Shelly makes the first shot, we move on to the second shot.
- The probability of making the second shot, given that she made the first shot, is 80% (0.8).
- If Shelly makes both the first and second shots, we move on to the third shot.
- The probability of making the third shot, given that she made the first two shots, is 90% (0.9).
- The total probability of Shelly making all three shots in this scenario is 0.6 * 0.8 * 0.9 = 0.432.
In this scenario, Shelly will score 3 points if she makes all three shots.
Second scenario: Shelly misses the first shot but makes the second and third shots.
- The probability of missing the first shot is 40% (1 - 0.6).
- If Shelly misses the first shot, we move directly to the second shot.
- The probability of making the second shot, given that she missed the first shot, is still 80% (0.8).
- If she makes the second shot, we move on to the third shot.
- The probability of making the third shot, given that she missed the first shot but made the second shot, is still 90% (0.9).
- The total probability of Shelly making the second and third shots but missing the first shot in this scenario is 0.4 * 0.8 * 0.9 = 0.288.
In this scenario, Shelly will score 2 points if she makes the last two shots.
Third scenario: Shelly misses the first and second shots but makes the third shot.
- The probability of missing the first shot is already mentioned as 40% (1 - 0.6).
- If Shelly misses the first shot, we move directly to the second shot.
- The probability of missing the second shot, given that she missed the first shot, is 20% (1 - 0.8).
- If she misses the second shot, we move on to the third shot.
- The probability of making the third shot, given that she missed the first two shots, is still 90% (0.9).
- The total probability of Shelly making only the third shot but missing the first two shots in this scenario is 0.4 * 0.2 * 0.9 = 0.072.
In this scenario, Shelly will score 1 point if she makes only the last shot.
Now, we need to calculate the expected score by considering the probabilities of each scenario and the points scored in each scenario:
Expected score = (Probability of scenario 1) * (Points in scenario 1) +
(Probability of scenario 2) * (Points in scenario 2) +
(Probability of scenario 3) * (Points in scenario 3)
Expected score = (0.432 * 3) + (0.288 * 2) + (0.072 * 1) = 1.296 + 0.576 + 0.072 = 1.944
Therefore, Shelly is most likely to score 1.944 points in a one-and-one-and-one situation on Mars.