Carlos plays college soccer. He makes a goal 65% of the time he shoots. Carlos is going to attempt two goals in a row in the next game. A = the event Carlos is successful on his first attempt. P(A) = 0.65. B = the event Carlos is successful on his second attempt. P(B) = 0.65. Carlos tends to shoot in streaks. The probability that he makes the second goal given that he made the first goal P(B/A ) is 0.90. What is the probability that he makes both goals P (A and B)?
Question
Carlos plays college soccer. He makes a goal 65% of the time he shoots. Carlos is going to attempt two goals in a row in the next game. A = the event Carlos is successful on his first attempt. P(A) = 0.65. B = the event Carlos is successful on his second attempt. P(B) = 0.65. Carlos tends to shoot in streaks. The probability that he makes the second goal given that he made the first goal P(B/A ) is 0.90. What is the probability that he makes both goals P (A and B)?
Solution
The probability of two independent events both occurring is the product of their individual probabilities. However, in this case, the events are not independent because the probability of the second event (Carlos making the second goal) is affected by the outcome of the first event (Carlos making the first goal). This is known as conditional probability.
The formula for the probability of two dependent events both occurring is P(A and B) = P(A) * P(B|A).
Here, P(A) is the probability that Carlos makes the first goal, which is 0.65. P(B|A) is the probability that Carlos makes the second goal given that he made the first goal, which is 0.90.
So, the probability that Carlos makes both goals is P(A and B) = P(A) * P(B|A) = 0.65 * 0.90 = 0.585 or 58.5%.
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