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Question
Solve the following problem :
In a large school, 80% of the students like mathematics. A visitor asks each of 4 students, selected at random, whether they like mathematics.
Find the probability that the visitor obtains the answer yes from at least 3 students.
Solution
Let X denote the number of pupils who like mathematics.
P(pupils like mathematics) = p = `(8)/(100) = (4)/(5)` ...[Given]
q = 1 – p = `1 - (4)/(5) = (1)/(5)`
Given, n = 4
∴ X ~ B`(4, 4/5)`
The p.m.f. of X is given by
P(X = x) = `""^4"C"x 4/5^x (1/5)^(4 - x), x` = 0, 1, ...,4
P(the visitor obtains the answer yes from at least 3 students)
= P(X ≥ 3)
= P(X = 3 or X = 4)
= P(X = 3) + P(X = 4)
= `""^4"C"_3 (4/5)^3 (1/5)^1 + (256)/(5^4)` ...[From (i)]
= `(4^4)/(5^4) + (256)/(5^4)`
= `(256)/(5^4) + (256)/(5^4)`
= `(512)/(5^4)`.
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