Question

For each lecture the professor chooses between white, yellow, orange, and purple chalk, independently of previous choices. Each day she chooses white chalk with probability 0.4, yellow chalk with probability 0.3, orange chalk with probability 0.2, and purple chalk with probability 0.1.

(a) Over the next 10 days, let W and Y be the number of days she chooses white chalk and yellow chalk, respectively. Are W and Y independent? Why or why not?

(b) What is the probability that over the next 10 days she will choose white chalk 5 times, yellow chalk 3 times?

Answer 1

a.

W and Y are not independent

explanation :

let us say W = 1 then Y can be from 0 to 9

but if W = 2 then Y=9 is not possible therefore the value of W does influence the value of Y

therefore,

they are not independent

b.

given :

p(white chalk) = 0.4

p(yellow chalk) = 0.3

p(not yellow, not white) = p(orange) + p(purple) = 0.3

Selecting 5 days from 10 when she uses white chalk = 10C5 and probability of using white chalk = p(white chalk)^5 = 0.4^5

Selecting 3 days from remaining 5 when she uses yellow chalk, 5C3 and probability of using yellow chalk is p(yellow chalk)^3 = 0.3^3

for remaining 2 days she must use chalk that is not yellow and not white,

probability for not using white, yellow for 2 days = p(not yellow, not white)^2 = (0.3)^2

Required probability = (10C5*0.4^5) * (5C3*0.3^3) * (2C2*0.3^2) = 0.0627

P.S. (please upvote if you find the answer satisfactory)