Question

10 digit phone number (reps 0-9)

All possible combos?

probability of 123-456-7890 being dialed?

Has at least one 5?

Answer 1

P(A) = n(E)/n(S)

Where P(A) is the probability of an event A

n(E) is the number of favorable outcomes

n(S) is the total number of events in the sample space.      

a)

A 10-digit phone number can be formed using 0 to 9 digits. It means every number has 10 options of digits to fill.

Total number of phone numbers = 10*10*10*10*10*10*10*10*10*10 = 10,0000000000.

b)

There will be just one case of 123-456-7890.

Hence, the probability will be 1/10,0000000000.

c)

Atleast one 5 means one or more than one 5s.

We can calculate the probability of having no 5s and then we can subtract it from 1 to get the probability of at least one 5.

Number of cases when there is no 5s = 9*9*9*9*9*9*9*9*9*9 = 9**10

Probability(at least one 5) = 1 - Probability(getting no 5s)

Probability(at least one 5) = 1 - 1/(9**10)

The probability is (9**10 - 1) / 9**10.