Calculating the Probability of Exactly One Brother Passing an Exam

In the following scenario, we examine the probability of exactly one out of two brothers passing a critical exam. This problem is relevant for understanding the intricacies of probability and can be a useful example for students and professionals alike when dealing with similar scenarios.

Understanding the Problem

We have two brothers who appear for an examination. The probability of each brother being selected is given as follows:

Probability of brother A being selected (pA) 1/4 Probability of brother B being selected (pB) 2/9

The complementary probabilities, or the chances of them not being selected, are:

Probability of brother A not being selected 1 - 1/4 3/4 Probability of brother B not being selected 1 - 2/9 7/9

Probability of Exactly One Sibling Passing the Exam

To find the probability that exactly one brother passes the exam, we need to consider the following scenarios:

Brother A passes and brother B does not. Brother A does not pass and brother B passes.

The formula for the probability of exactly one of them getting selected is:

P(exactly one gets selected) (P(A passes and B fails) P(A fails and B passes))

Calculating these probabilities:

P(A passes and B fails)  (1/4) * (7/9)  7/36
P(A fails and B passes)  (3/4) * (2/9)  6/36

The combined probability is:

P(exactly one gets selected) 7/36 6/36 13/36

Alternative Scenario with Different Pass Rates

Now, consider a more complex situation where the pass rates are different. Let's say the probability that A passes is 3/5, and the probability that B passes is 5/8. The probability that A fails and B passes can be calculated as:

Probability that A fails 1 - 3/5 2/5

Probability that B passes 5/8

Hence, the probability that only one will pass is:

P(only one passes) (3/5) * (3/8) (2/5) * (5/8)

Therefore:

P(only one passes) (9/40) (10/40) 19/40

Conclusion

This detailed analysis helps to grasp the concept of probability in scenarios where multiple independent events can occur. It is a valuable tool for students and professionals in fields such as statistics, data analysis, and decision-making processes. Understanding these principles can assist in making informed decisions and predicting outcomes in various real-world scenarios.

Keywords: probability, exam, siblings, selection