A bag consists of 10 balls each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?

#### Solution

Let X denote the number of balls marked with the digit 0 among the 4 balls drawn.

Since the balls are drawn with replacement, the trials are Bernoulli trials.

X has a binomial distribution with n = 4 and p =`1/10`

and q = 1 - p = `1 - 1/10 = 9/10`

The p.m.f. of X is given by

P(X = x) = `"^nC_x p^x q^(n-x)`

i.e. p(x) = `"^4C_x (1/10)^x (9/10)^(4-x)`, x = 0, 1, ...,4

P(none of the ball marked with digit 0) = P(X = 0)

= p(0) = `"^4C_x (1/10)^0 (9/10)^(4 - 0)`

`= 1xx1 xx (9/10)^4 = (9/10)^4`

Hence, the probability that none of the bulb marked with digit o is `(9/10)^4`.