Question

Among 28 professors of a certain department, 18 drive foreign cars and 10 drive local made cars. If 5 of these professors are selected at random, what is the probability that atleast 3 of them drive foreign cars?

Answer 1

Let p be the probability of foreign cars driven by the professors

p = `18/28 = 9/14`

Let q be the probability of local made cars driv¬en by the professors

q = `10/28 = 5/14`

and n = 5

The binomial distribution p(x = x) = nc_xp^xq^n-x 

`"p"(x = x) = 5"c"_x (9/14)^x (5/14)^(5 - x)`

p(altleast 3 of them drive foreign cars) = p(x ≥ 3)

p(x ≥ 3) = p(x = 3) + p(x = 4)+ p(x = 5)

= `5"c"_3 (9/14)^3(5/1)^(5-3) + 5"c" (9/14)^4 (5/14)^(5 - 4) + 5"c"_5 (9/14)^5 (5/14)^(5 - 5)`

= `5"c"_2 (9/14)^3 (5/14)^2 + 5"c"_1 (9/14)^4 (5/14)^0 + 5"c"_0 (9/14)^5 (5/14)^0`

= `((5 xx 4)/(1 xx 2)) xx ((9^3 xx 5^2)/(14^5)) + 5 xx ((9^4 xx 5^1)/14) + (9^5/14^5)`

= `((10 xx 9^3 xx 5^2) + (5 xx 9^4 xx 5) + (9^5))/14^5`

= `((10 xx 729 xx 25) + (5 xx 9^4 xx 5) - (9^5))/537824`

= `405324/537824`  .......`{"nc"_"r" = "nc"_("n" - "r")}`

= 0.7536