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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.

User Bopjesvla
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Answer:

(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

User Feniix
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