Question

A five​-digit number starts with a number between 4​-9 in the first​ position, with no restrictions on the remaining 4 digits. a) Find the probability that a​ randomly-chosen phone number contains all different digits. b) Find the probability that a​ randomly-chosen phone number contains at least one repeated digit. ​a) Write an expression that models the probability. Select the correct choice below and fill in the answer​ box(es) within your choice.

Answer 1

(a)
`Pr = 0.3024`

(b)
`Pr = 0.6976`

(c)
`Pr = (^9P_(n-1))/(10^(n-1))`

Explanation:

Given

`Start = \{5,6,7,8\}` i.e. between 4 and 9

`n(Start) =4`

`Digits = 5`

Solving (a): Probability that each of the 5 digit are different

Since there is no restriction;

The total possible selection is as follows:

`First\ digit = 4` (i.e. any of the 4 start digits)

`Second\ digit = 10\\` (i.e. any of the 10 digits 0 - 9)

`Third\ digit = 10` (i.e. any of the 10 digits 0 - 9)

`Fourth\ digit = 10` (i.e. any of the 10 digits 0 - 9)

`Fifth\ digit = 10` (i.e. any of the 10 digits 0 - 9)

So, the total is:

`Total = 4 * 10 * 10 * 10 * 10`

`Total = 40000`

For selection that all digits are different, the selection is:

`First\ digit = 4` (i.e. any of the 4 start digits)

`Second\ digit = 9` (i.e. any of the remaining 9)

`Third\ digit = 8` (i.e. any of the remaining 8)

`Fourth\ digit = 7` (i.e. any of the remaining 7)

`Fifth\ digit = 6` (i.e. any of the remaining 6)

So:

`Selection =4 * 9 * 8 * 7 * 6`

`Selection =12096`

So, the probability is:

`Pr = (Selection)/(Total)`

`Pr = (12096)/(40000)`

`Pr = 0.3024`

Solving (b): At least 1 repeated digit

The probability calculated in (a) is the all digits are different i.e. P(None)

So, using laws of complement

We have:

`P(At\ least\ 1) = 1 - P(None)`

So, we have:

`Pr= 1 - 0.3024`

`Pr = 0.6976`

Solving (c): An expression to model the probability.

Using (a) as a point of reference, we have;

`Pr = (Selection)/(Total)`

Where

`Selection =4 * 9 * 8 * 7 * 6` ---- for selection of 5 i.e. n = 5

`Total = 4 * 10 * 10 * 10 * 10`

`Selection =4 * 9 * 8 * 7 * 6`

This can be rewritten as:

`Selection = 4 * ^9P_4`

4 can be expressed as: 5 - 1

So, we have:

`Selection = (5-1) *^9P_(5-1)`

Substitute n for 5

`Selection = (n-1) *^9P_(n-1)`

`Selection = (n-1)^9P_(n-1)`

`Total = 4 * 10 * 10 * 10 * 10`

This can be rewritten as:

`Total = 4 * 10^4`

`Total = (5-1) * 10^(5-1)`

`Total = (n-1) * 10^(n-1)`

`Total = (n-1) 10^(n-1)`

So, the expression is:

`Pr = ((n-1)^9P_(n-1))/((n-1)10^(n-1))`

`Pr = (^9P_(n-1))/(10^(n-1))`

Where n represents the digit number