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 3/5 of the games and goalie B plays 2/5 of the games then:
1. Determine the probability that a save has been made by either goalie A or goalie B.
2. If a save has been made, what is the probability that is was goalie A?
3.If a goal was let in, what is the probability it was goalie B?
THANKS!!
To determine the answers to these questions, we can use basic probability calculations. Let's break down each question and calculate the probabilities step by step:
1. Determine the probability that a save has been made by either goalie A or goalie B:
First, we need to calculate the overall probability that a save has been made by either goalie A or goalie B. To do this, we'll calculate the weighted sum of the individual probabilities, taking into account the number of games each goalie plays:
Probability(A or B) = (Probability(A) x Fraction of games played by A) + (Probability(B) x Fraction of games played by B)
Probability(A) = 80% = 0.8
Probability(B) = 75% = 0.75
Fraction of games played by A = 3/5 = 0.6
Fraction of games played by B = 2/5 = 0.4
Probability(A or B) = (0.8 x 0.6) + (0.75 x 0.4) = 0.48 + 0.3 = 0.78
Therefore, the probability that a save has been made by either goalie A or goalie B is 0.78 or 78%.
2. If a save has been made, what is the probability that it was goalie A:
We need to find the conditional probability of goalie A making a save given that a save has been made. This can be calculated using Bayes' theorem:
Probability(A|save) = (Probability(A) x Fraction of games played by A) / (Probability(A or B))
Using the values from the previous calculation:
Probability(A|save) = (0.8 x 0.6) / 0.78 = 0.48 / 0.78
Probability(A|save) ≈ 0.6154
Therefore, if a save has been made, the probability that it was goalie A is approximately 0.6154 or 61.54%.
3. If a goal was let in, what is the probability it was goalie B:
To find the probability that goalie B let in a goal, we can subtract the conditional probability of goalie A making a save from 1 (since if a goal was let in, it couldn't have been saved by goalie A):
Probability(B|goal) = 1 - Probability(A|save)
Probability(B|goal) = 1 - 0.6154
Probability(B|goal) ≈ 0.3846
Therefore, if a goal was let in, the probability that it was goalie B is approximately 0.3846 or 38.46%.
I hope this explanation helps! Let me know if you have any further questions.