Question

What is the probability that a five-digit number has at least one zero in it? (a) 0.296 (b) 0.314 (c) 0.344 (d) 0.412 (e) 0.442

Answer 1

Correct option (c) 0.344.

Step-by-step explanation:

The five-digit number can be formed by any of the 10 numbers from the set:

S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

The total number of 5-digit number is,

_, _, _, _, _

The first digit cannot be 0.

So the first place can be filled in 9 ways.

The remaining 4 places can be filled in 10 ways each.

Total number of 5-digit number = 9 × 10 × 10 × 10 × 10 = 90,000.

The total number of 5-digit number not having 0 is:

All the five places can be filled by any of the 9 numbers from the set,

S = {1, 2, 3, 4, 5, 6, 7, 8, 9}

Total number of 5-digit number without 0 is, 9 × 9 × 9 × 9 × 9 = 59,049.

Compute the probability that a five-digit number has at least one zero in it as follows:

P (At least one 0) = 1 - P (No 0)

                             $=1-\frac{59049}{90000}$

                             $=\frac{90000-59049}{90000}$

                             $=\frac{30951}{90000}$

                             $=0.3439\\\approx0.344$

Thus, the probability that a five-digit number has at least one zero in it is 0.344.