A bus company has a large number of buses - OCR - GCSE Maths - Question 15 - 2020 - Paper 6

Let P(A') be the probability of selecting a bus younger than 10 years. Therefore,

$P(A') = 1 - P(A) = 1 - 0.25 = 0.75$

The probability of this bus starting first time given that it is younger than 10 years, P(B|A'), is 0.65.

Thus, the contribution to a successful start from buses younger than 10 years is:

$P(A') \cdot P(B|A') = 0.75 \cdot 0.65 = 0.4875$

To find the total probability that the bus starts first time, we add the contributions from both age groups:

$P(B) = P(A) \cdot P(B|A) + P(A') \cdot P(B|A')$

Substituting the calculated values:

$P(B) = 0.075 + 0.4875 = 0.5625$

Thus, the probability that the bus starts first time is (0.5625) or 56.25%.