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Mr. A either walks to school or take bus to school everyday. The probability that he takes a bus to school is $1 / 4 .$ If he takes a bus, the probability that he will be late is $2 / 3 .$ If he walks to school, the probability that he will be late is $1 / 3 .$ The probability that Mr. A will be on time for at least one out of two consecutive days is $\frac{p}{q},$ where p and q are co-prime , the value of $(q - p)$ is

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answer is 25.

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Detailed Solution

A: Mr. A reaches late

$B_{1} :$ A goes to school by walking

$B_{2} :$ A takes bus to school.

E: A will be on time for atleast one out of 2 consecutive days.

$P (B_{1}) = 3 / 4$

$P (B_{2}) = 1 / 4$

$P (A / B_{1}) = 1 / 3$

$P (A / B_{2}) = 2 / 3$

$∴ P (A) = P (B_{1} \cap A) + P (B_{2} \cap A) = \frac{3}{4} \times \frac{1}{3} + \frac{1}{4} \times \frac{2}{3} = \frac{5}{12}$

$∴ P (E) = 1 - P (A \cap A) = 1 - \frac{5}{12} \times \frac{5}{12} = \frac{119}{144} \equiv \frac{p}{q} \Rightarrow q - p = 144 - 119 = 25$

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