Answered

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 :

MrRoyal

Answer:

(a) [tex]Pr = 0.3024[/tex]

(b) [tex]Pr = 0.6976[/tex]

(c) [tex]Pr = \frac{^9P_{n-1}}{10^{n-1}}[/tex]

Step-by-step explanation:

Given

[tex]Start = \{5,6,7,8\}[/tex] i.e. between 4 and 9

[tex]n(Start) =4[/tex]

[tex]Digits = 5[/tex]

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

Since there is no restriction;

The total possible selection is as follows:

[tex]First\ digit = 4[/tex] (i.e. any of the 4 start digits)

[tex]Second\ digit = 10\\[/tex] (i.e. any of the 10 digits 0 - 9)

[tex]Third\ digit = 10[/tex] (i.e. any of the 10 digits 0 - 9)

[tex]Fourth\ digit = 10[/tex] (i.e. any of the 10 digits 0 - 9)

[tex]Fifth\ digit = 10[/tex] (i.e. any of the 10 digits 0 - 9)

So, the total is:

[tex]Total = 4 * 10 * 10 * 10 * 10[/tex]

[tex]Total = 40000[/tex]

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

[tex]First\ digit = 4[/tex] (i.e. any of the 4 start digits)

[tex]Second\ digit = 9[/tex] (i.e. any of the remaining 9)

[tex]Third\ digit = 8[/tex] (i.e. any of the remaining 8)

[tex]Fourth\ digit = 7[/tex] (i.e. any of the remaining 7)

[tex]Fifth\ digit = 6[/tex] (i.e. any of the remaining 6)

So:

[tex]Selection =4 * 9 * 8 * 7 * 6[/tex]

[tex]Selection =12096[/tex]

So, the probability is:

[tex]Pr = \frac{Selection}{Total}[/tex]

[tex]Pr = \frac{12096}{40000}[/tex]

[tex]Pr = 0.3024[/tex]

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:

[tex]P(At\ least\ 1) = 1 - P(None)[/tex]

So, we have:

[tex]Pr= 1 - 0.3024[/tex]

[tex]Pr = 0.6976[/tex]

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

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

[tex]Pr = \frac{Selection}{Total}[/tex]

Where

[tex]Selection =4 * 9 * 8 * 7 * 6[/tex] ---- for selection of 5 i.e. n = 5

[tex]Total = 4 * 10 * 10 * 10 * 10[/tex]

[tex]Selection =4 * 9 * 8 * 7 * 6[/tex]

This can be rewritten as:

[tex]Selection = 4 * ^9P_4[/tex]

4 can be expressed as: 5 - 1

So, we have:

[tex]Selection = (5-1) *^9P_{5-1}[/tex]

Substitute n for 5

[tex]Selection = (n-1) *^9P_{n-1}[/tex]

[tex]Selection = (n-1)^9P_{n-1}[/tex]

[tex]Total = 4 * 10 * 10 * 10 * 10[/tex]

This can be rewritten as:

[tex]Total = 4 * 10^4[/tex]

[tex]Total = (5-1) * 10^{5-1}[/tex]

[tex]Total = (n-1) * 10^{n-1}[/tex]

[tex]Total = (n-1) 10^{n-1}[/tex]

So, the expression is:

[tex]Pr = \frac{(n-1)^9P_{n-1}}{(n-1)10^{n-1}}[/tex]

[tex]Pr = \frac{^9P_{n-1}}{10^{n-1}}[/tex]

Where n represents the digit number

Other Questions