Understanding the Examination Probability Problem

This question asks for the probability that either student A or student B fails an examination. We are given the individual probabilities of failure for each student.

Let's define the events:

• Event A: Student A fails the examination.
• Event B: Student B fails the examination.

We are given the following probabilities:

• The probability of A failing is $P(A_{fail}) = 0.2$.
• The probability of B failing is $P(B_{fail}) = 0.3$.

The question asks for the probability that "either A or B fails". Based on the provided answer options, this phrase is interpreted as the probability that *exactly one* of the students fails the examination. This can occur in two distinct scenarios:

1. Scenario 1: Student A fails, AND Student B passes.
2. Scenario 2: Student A passes, AND Student B fails.

To calculate these, we first need the probabilities of each student passing:

• The probability of A passing is $P(A_{pass}) = 1 - P(A_{fail}) = 1 - 0.2 = 0.8$.
• The probability of B passing is $P(B_{pass}) = 1 - P(B_{fail}) = 1 - 0.3 = 0.7$.

We assume that the events of A failing and B failing are independent. This is a common assumption in such problems unless stated otherwise.

Calculating Probability of Exactly One Failure

Now, we calculate the probability for each scenario:

Scenario 1: A Fails and B Passes

Using the independence assumption, the probability of this scenario is:

$$ P(\text{A fails AND B passes}) = P(A_{fail}) \times P(B_{pass}) $$

$$ P(\text{A fails AND B passes}) = 0.2 \times 0.7 = 0.14 $$

Scenario 2: A Passes and B Fails

Similarly, the probability of this scenario is:

$$ P(\text{A passes AND B fails}) = P(A_{pass}) \times P(B_{fail}) $$

$$ P(\text{A passes AND B fails}) = 0.8 \times 0.3 = 0.24 $$

Determining the Final Probability

Since Scenario 1 and Scenario 2 are mutually exclusive events (they cannot happen at the same time), the total probability that *either* A or B fails (meaning exactly one fails) is the sum of their probabilities:

$$ P(\text{Exactly one fails}) = P(\text{A fails AND B passes}) + P(\text{A passes AND B fails}) $$

$$ P(\text{Exactly one fails}) = 0.14 + 0.24 $$

$$ P(\text{Exactly one fails}) = 0.38 $$

Therefore, the probability that either A or B fails the examination is 0.38.