19. A high school soccer team has two goalies. Goalie A makes the save 80% of the time. Goalie B makes the save 75% of the time. If goalie A plays 35 of the games and goalie B plays 25 of the games then:

a) Determine the probability that a save has been made by either goalie A or goalie B.

b) If a save has been made, what is the probability that is was goalie A?

c) If a goal was let in, what is the probability it was goalie B?

(a) (35/60)*0.8 + (25/60)*0.75 = 0.779

(b) Assume N shots on goal per game, and 60 total games. Probability of save by A
= (0.8)*N*35/[0.8*N*35 + 0.75*N*25]
= 28/(28 + 18.75) = 0.599

(c) Your turn

To solve these probability questions, we can use the concepts of probability and conditional probability. Let's break down the problem and calculate the probabilities step by step:

a) Determine the probability that a save has been made by either goalie A or goalie B.

Goalie A plays 35 games, and the probability of making a save is 80%. Therefore, the number of saves made by goalie A can be calculated as 0.8 * 35 = 28 saves.

Similarly, goalie B plays 25 games, and the probability of making a save is 75%. So, the number of saves made by goalie B can be calculated as 0.75 * 25 = 18.75 saves. Since we cannot have a fractional number of saves, we round it to the nearest whole number, which is 19.

Now, to calculate the probability that a save has been made by either goalie A or goalie B, we add the number of saves made by each goalie and divide it by the total number of games played. The total number of saves is 28 + 19 = 47, and the total number of games played is 35 + 25 = 60.

Therefore, the probability (P) that a save has been made by either goalie A or goalie B is:

P = (Total number of saves) / (Total number of games played)
P = 47 / 60
P ≈ 0.7833 or 78.33%

So, the probability that a save has been made by either goalie A or goalie B is approximately 0.7833 or 78.33%.

b) If a save has been made, what is the probability that it was goalie A?

To calculate this, we need to use conditional probability. The probability that it was goalie A given that a save has been made can be represented as:

P(Goalie A|Save) = P(Goalie A and Save) / P(Save)

The P(Goalie A and Save) is the number of saves made by goalie A, which is 28, divided by the total number of games played, which is 60. So, P(Goalie A and Save) = 28 / 60.

The P(Save) is the probability that a save has been made, which we calculated as 0.7833 or 78.33%.

Now we can calculate the probability that it was goalie A given that a save has been made:

P(Goalie A|Save) = (28 / 60) / 0.7833
P(Goalie A|Save) ≈ 0.5991 or 59.91%

So, if a save has been made, the probability that it was made by goalie A is approximately 0.5991 or 59.91%.

c) If a goal was let in, what is the probability it was goalie B?

To solve this, we need to use the concept of complementary probability. If a save was not made, it means a goal was let in. So, the probability that the goal was let in by goalie B can be represented as:

P(Goalie B|Goal) = 1 - P(Goalie A|Save)

We already calculated P(Goalie A|Save) as 0.5991 or 59.91%. Therefore:

P(Goalie B|Goal) = 1 - 0.5991
P(Goalie B|Goal) ≈ 0.4009 or 40.09%

So, if a goal was let in, the probability that it was let in by goalie B is approximately 0.4009 or 40.09%.