Analyzing Carlos’s Soccer Skills: Probability of Scoring Goals

QUESTION

1. Carlos plays college soccer.  Carlos is going to attempt two goals in a row.  P(A)=0.65 which is the probability that he will make a goal on the first attempt.  P(B)=0.65 which is the probability that he will make a goal on the second attempt.  The probability that he makes both goals = 0.585.
2. What is the probability that he makes either the first goal or the second goal?
3. What is the probability that he makes the second goal given that he made the first goal?

ANSWER

Analyzing Carlos’s Soccer Skills: Probability of Scoring Goals

Introduction

In this analysis, we will explore the probabilities associated with Carlos, a college soccer player, attempting two consecutive goals. We are given the individual probabilities of Carlos making the first and second goals, as well as the probability of him making both goals. We will then determine the probability of Carlos scoring either the first goal or the second goal. Furthermore, we will calculate the conditional probability of Carlos making the second goal, given that he successfully made the first goal.

Probability of Scoring Either the First or Second Goal

To find the probability of Carlos making either the first or the second goal, we need to consider the concept of mutually exclusive events. Since Carlos can only make one goal at a time (assuming no rebounds or multiple shots), the events of scoring the first and the second goal are mutually exclusive.

Let’s denote the probability of making the first goal as P(A) = 0.65, and the probability of making the second goal as P(B) = 0.65. To calculate the probability of making either the first or the second goal, we can use the principle of addition.

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Given that P(A) = P(B) = 0.65 and the probability of making both goals, P(A ∩ B) = 0.585, we can substitute these values into the formula:

P(A ∪ B) = 0.65 + 0.65 – 0.585 = 1.315 – 0.585 = 0.73

Therefore, the probability that Carlos makes either the first goal or the second goal is 0.73.

Conditional Probability of Making the Second Goal Given the First Goal

The conditional probability measures the likelihood of an event occurring given that another event has already occurred. In this case, we want to calculate the probability of Carlos making the second goal, given that he successfully made the first goal.

The conditional probability formula is as follows:

P(B|A) = P(A ∩ B) / P(A)

Substituting the given values, we have:

P(B|A) = 0.585 / 0.65 = 0.9

Therefore, the probability that Carlos makes the second goal given that he made the first goal is 0.9.

Conclusion

In this analysis, we examined the probabilities associated with Carlos’s soccer skills, specifically his chances of making two consecutive goals. We determined that the probability of Carlos making either the first goal or the second goal is 0.73, based on the individual probabilities of scoring each goal. Additionally, the probability of Carlos making the second goal, given that he made the first goal, is 0.9. These probabilities provide insights into Carlos’s performance and can be useful for understanding his goal-scoring abilities on the field.