Question

6. In the Big Four lottery game, a four-digit number is drawn at random, where each digit is selected from the set {0, 1, 2,..., 9}. Let Z = the number of 7’s in the number. Determine the probability mass function of Z.

Answer 1

In a four digit number there could be 0, 1, 2, 3, 4, number of 7's.

Now, let the probability of drawing one 7 from a set {0, 1, 2,..., 9} be p.

Let Z: the number of 7's in the number

Z can take any value from the set {0, 1, 2, 3, 4}. So, the pmf of Z is

To explain this let us find probability of two 7's in the number i.e., P(Z=2). For this we have to multiply probability of getting a 7 two times and probability of not getting a 7 remaining two times (as it is a four digit number). Now, we can get two 7's in many ways like:

7,7,*,*

7,*,7,*

7,*,*,7

*,7,*,7

*,*,7,7

*,7,7,*

There are total of 6 cases. We can find this out with the help of combination. ⁴C₂ means getting 2 7's out of 4 times. So, the probability is: