A candidate takes a two-stage driving examination consisting of a theoretical test followe… — JAMB UTME

A candidate takes a two-stage driving examination consisting of a theoretical test followed by a practical test. The probability that the candidate passes the theoretical test is $0.75$ . If the candidate passes the theoretical test, the probability of passing the practical test is $0.80$ . However, if the candidate fails the theoretical test, the probability of passing the practical test is $0.20$ . What is the probability that the candidate passes exactly one of the two tests?

Answer: 0.2

Answer

The probability that the candidate passes exactly one of the two tests is 0.2.

To pass exactly one test, the candidate must either pass theory and fail practical (

0.75 \times 0.20 = 0.15

) or fail theory and pass practical (

0.25 \times 0.20 = 0.05

). Since these two outcomes cannot happen at the same time, their probabilities are added:

0.15 + 0.05 = 0.20

Step-by-Step Solution

Calculate the probability of passing the theoretical test and failing the practical test.

P(\text{Theory} \cap \text{Practical}') = 0.75 \times (1 - 0.80) = 0.15

The probability of failing practical given passing theory is

1 - 0.80 = 0.20

Calculate the probability of failing the theoretical test and passing the practical test.

P(\text{Theory}' \cap \text{Practical}) = (1 - 0.75) \times 0.20 = 0.05

The probability of failing theory is

1 - 0.75 = 0.25

Sum the probabilities of the two mutually exclusive outcomes.

0.15 + 0.05 = 0.20

Passing exactly one test is the union of two disjoint compound events.

Key Concept

Compound probability laws and conditional independence structure in sequential events

Estimated Time:1m 30s

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