Question

For segments (a) and (b), use the information below.

The personal identification numbers (PINs) for ATMs usually consist of four digits, chosen from 0, 1, 2, ….., 9. Suppose you notice that most of the PINs you hold have at least one “1,” which makes you wonder if the issuers of these numbers include many ones so that users will remember them more easily.

                Assume PIN numbers of 4 digits are assigned randomly.

1. How many unique PIN numbers are possible? (Note: Please make sure to write out on your Excel sheet what you entered into to a cell to compute your answer.)

2. Compute the probability that a PIN assigned at random has at least one “1.” Round to 4 digits. (Hint: might taking the complement help here?)

(Note: again, please type out next to your answer the commands used to compute it.)              

3. Social Security numbers have 9 digits. How many unique social security numbers are possible?

4. Suppose the government decided to change the format of social security numbers, so that the first three digits are letters instead. And that any of these letters, as well as any of the remaining 6 digits could not repeated. How any unique social security numbers would then be possible? (Remember, the alphabet has 26 letters! And that we do not assume replacement, since both letters and numbers cannot be repeated.)

		Answers ê	work
(a)	Pin numbers?
(b)	P(at least one "1")?
( c)	Unique SSNs?
(d)	Unique SSNs?

Answer 1

Solution

Back-up Theory

Probability of an event E, denoted by P(E) = n/N …………………………..........................................……………………(1)

where

n = n(E) = Number of outcomes/cases/possibilities favorable to the event E and

N = n(S) = Total number all possible outcomes/cases/possibilities.

Now, to work out the solution,

Part (a)

First of the four digits can be any one of 10 digits, 0, 1, 2, ......., 9 in 10 ways.

Since there is no restriction of repetition, the second digit can also be any one of 10 digits, 0, 1, 2, ......., 9 in 10 ways.

Similarly, the third and fourth digits can also be any one of 10 digits, 0, 1, 2, ......., 9 in 10 ways.

Thus, total number of unique PIN numbers possible = 10⁴

= 10000 Answer 1

Part (b)

By the same arguments as in Part (a), total number of possible unique PIN numbers in which ‘1’ does not figure at any position = 9⁴

= 6561

[using Excel function: Math & Trig POWER with 9 against Number and 4 against Power]

So, by complement, total number of possible unique PIN numbers in which ‘1’ figures at least once

= 10⁴ – 9⁴

= 3439

This represents n vide (1) and N = 10000. Hence,

the probability that a PIN assigned at random has at least one “1.” = 3439/10000 = 0.3439 Answer 2

Part (c)

Social Security numbers have 9 digits.

Following the analysis identically as above, replacing 4 by 9,

total number of unique social security numbers possible = 10⁹ = 1000000000 Answer 3

[using Excel function: Math & Trig POWER with 10 against Number and 9 against Power]

Part (d)

Here, we have three letters followed by six numerals (digits) with the additional condition that no letter and no numeral can repeat in the same SSN.

The first letter can be any one of 26 letters in 26 ways. Since repetition is not allowed, the second letter can be only any one of the remaining 25 letters in 25 ways and the third letter can also be only any one of the remaining 24 letters in 24 ways.

Thus, three letters can be formed in 26 x 25 x 24 = 15600 ways.

By the same arguments, the 6 numerals can be formed in 10 x 9 x 8 x 7 x 6 x 5 = 151200 ways.

Combining the two, total number of possible unique Social Security Numbers = 15600 x 151200

= 2358720000 Answer 4

DONE