Probability

Question

A computer is programmed to produce at random a single digit from the list 0,1,2,3,4,5,6,7,8,9. The program runs twenty times. Let Y be the number of zeros that occur.

a) State the distribution of Y and give its parameters 

b) Calculate P(Y<3). 

Answer 1

Hi Faith,

 

a) Distribution of Y will be binomial distribution because there are only two possible outcomes, i.e., either getting a zero or a different number. The parameters of binomial distribution are n and p. n is the total number of events and p is the probability of success.

n = 20

p = 1/10

 

b) P(Y<3) means getting a 0 at least two times.

P(Y<=2) = P(Y=0) + P(Y=1) + P(Y=2)

In mathematical notation, we can write

P(Y<=2) = ∑(k=x to n)C(n k) p^kq^n-k 

x = 2, n = 20, p=1/10, q = 9/10

when x=2 (20 2)(1/10)^2(9/10)^20-2 = 0.285
when x=1 (20 1)(1/10)^1(9/10)^20-1 = 0.270
when x=0 (20 0)(1/10)^0(9/10)^20-0 = 0.122

Add them up, and you should get 0.677 or 68% (rounded to the nearest whole number).

You should notice that there is fairly high chance that the computer will generate a 0 at least two times out of 20 runs.