Suppose a 7-digit phone number is chosen at random.

Question

(a) What is the probability that the digits are distinct (i.e. that no digits repeat)? (b) What is the probability that all digits are odd? (c) What is the probability that it ends “309” and has distinct digits? (d) What is the probability that the phone number is a palindrome conditioned on not containing the digits 0 and 1?

Answer 1

Nazia,

The best way to approach these problems is to use the basic definition of probability and the multiplication counting rule. Probability of Event A = # of times Event A occurs / # of all events that could happen. The Multiplication counting rule says if A can happen m ways and B can happen n ways, the number ways that the combined situation AB can happen is n x m ways.

So for all these problems we first need to know how many possible 7 digit phone numbers are there to find out the denominator of the probability. So there are 10 possible digits in a phone number (0,1,2,3,4,5,6,7,8,9), so the total number of possible 7-digit telephone numbers is

10 x 10 x 10 x 10 x 10 x 10 x 10 = 10,000,000 possible numbers [technically, the first number probably can't be 0, so it really should be (9 x 10 x 10 x 10 x 10 x 10 x 10 = 9,000,000)]

Now you need to consider how to apply this counting rule to the numerators, but the strategy is to determine how many digits can possibly be in each of seven places in the phone number and then multiply those possibilities out.

I hope this helps. John