Question

Consider a series of digits selected at random and independently of each other. How many digits should be sampled in this way so that the probability that at least one 7 appears is at least i0

Answer 1

Define occurrence of 7 as a success, then probability of Success is 1/10=0.1.

Suppose that n digits are selected at random then define random variable X as number of 7^s in n digits.

X:0,1,,2…..n

Here X ~ B(n, p=0.1)

To find desired value of n we will use the following probabilistic equation.

    P[ X > 0] > 0.90

Or P[X=0] < 0.1

i.e. (1-p)ⁿ < 0.1

Taking log on both sides we will get

n Log(1-p) < Log(0.1)

n > Log(0.1)/Log(0.9) Inequality will change as both terms are negative.

n > 21.85 = 22.

Hence one needs to select 22 numbers so that 7 will come at least once with probability at least 0.9.