Question

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

If Yfollows the Binomial Distributions, state two assumptions for this.

i)Calculate the values of theparameters.

ii)Calculate P(Y<4)

Answer 1

A computer is programmed to produce at random a single digit from the list 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

The program is run 8 times.

Y is the number of zeroes that occur.

Now, the program can produce 10 digits (0 to 9) at every run.

So, the probability of getting a zero at any run is 1/10, or 0.1.

So, if Y be the number of zeroes that occur in 8 runs, then Y is a binomial random variable with parameters n=8 and p=0.1.

Question (i)

Y is a binomial random variable.

The assumptions for this are

(1) There are only two outcomes. Success is getting a zero, with probability 0.1, and failure is not getting a zero, with probability 0.9.

(2) The 8 runs that the program makes are independent of each other; the probability of success remains same in every run of the program.

Question (ii0)

Y follows binomial(8,0.1).

So, the values of the parameters are

n=8 and p=0.1.

Question (Iii)

We have to find P(Y<4).

The Probability Mass Function of Y is given by

Where, i can take integer values from 0 to 8.

So,

So, the probability Y<4, ie. P(Y<4) is 0.9950 approximately.