Question

Suppose again that the examination considered in Exercise 13 is taken by five students, and the first student to complete the examination finishes at 9:25 a.m. Determine the probability that at least one other student will complete the examination before 10:00 a.m.

Short answer

Probability that at least one other student will complete the examination before 10 a.m. is 0.8262

Step-by-step solution

Given information

The number of minutes required by any particular student to complete the examination has the exponential distribution with mean 80.

The first student to complete the examination finishes at 9.25 a.m.

Compute the probability

Let\({Y_2}\)be the time after one student to complete the examination.

Therefore,

\({Y_2}\) has the exponential distribution with parameter \(4\beta \)

Since, the mean of the distribution is 80.

\(\begin{array}{c}4\beta = 4 \times \frac{1}{{80}}\\ = \frac{4}{{80}}\\ = \frac{1}{{20}}\end{array}\)

Therefore,

Probability that at least one other student will complete the examination before 10 a.m. is\({\rm P}\left( {{Y_2} < 35} \right)\)

Since,

\({\rm P}\left( {x < X} \right) = 1 - F\left( x \right)\)

Therefore,

\(\begin{aligned}{}{\rm P}\left( {{Y_2} < 35} \right) &= 1 - {{\mathop{\rm e}\nolimits} ^{\left( { - 35 \times \frac{1}{{20}}} \right)}}\\ &= 1 - {{\mathop{\rm e}\nolimits} ^{\left( { - \frac{7}{4}} \right)}}\end{aligned}\)

\({\rm P}\left( {{Y_2} < 35} \right) = 0.8262\)

Hence, Probability that at least one other student will complete the examination before 10 a.m. is 0.8262.