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A state lottery game requires that you pick 6 different numbers from 1 to

​(a) Assuming that order is​ unimportant, what is the probability of picking all 6 numbers correctly to win the big​ prize?

​(b) What is the probability of picking exactly

55

of the 6 numbers​ correctly?

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Part 1

​(a) Assuming that order is​ unimportant, what is the probability of picking all 6 numbers correctly to win the big​ prize?

Part 2

Let S be a sample space of equally likely​ outcomes, and let event E be a subset of S. Then the probability that event E occurs is given by the formula

​P(E)equals=StartFraction n left parenthesis Upper E right parenthesis Over n left parenthesis Upper S right parenthesis EndFractionn(E)n(S).

Part 3

In this​ case, n(E) is the number of ways you can pick all 6 of the numbers correctly.

All 6 of the numbers can only be picked correctly 1 way.

Part 4

The sample space S is the set of all the ways you can pick 6 numbers. Since the numbers are selected without replacement and order does not​ matter,

​n(S)equals=​C(6565​,6).

There are

82 comma 598 comma 88082,598,880

ways that 6 numbers can be picked.

Part 5

Find the probability of picking all 6 numbers correctly. Round to eleven decimal places and then convert the number to scientific notation.

​P(E)

equals=

StartFraction n left parenthesis Upper E right parenthesis Over n left parenthesis Upper S right parenthesis EndFractionn(E)n(S)

equals=

StartFraction 1 Over 82 comma 598 comma 880 EndFraction182,598,880

equals=

1.211 times 10 Superscript negative 81.211×10−8

Part 6

​Thus, the probability of picking all 6 numbers correctly is

1.211 times 10 Superscript negative 81.211×10−8.

Part 7

​(b) What is the probability of picking exactly

55

of the 6 numbers​ correctly?

Part 8

Let

​n(E)equals=​"exactly

55

of the 6 numbers drawn were​ selected." The outcomes in E contain any 6​ numbers,

55

of which are from the 6 numbers drawn and the rest are not. Find the number of ways to select

55

numbers from the 6 numbers drawn.

Upper C left parenthesis 6 comma 5 right parenthesisC(6,5)

Part 9

From the total

6565

​numbers, there are only six matching the numbers drawn on the ticket.​ Hence, find the quantity of numbers that do not match the numbers drawn.

6565minus−6equals=5959

Part 10

Find the number of ways to select the remaining

11

numbernothing

from the

5959

numbers that are not drawn on the ticket.

Upper C left parenthesis 59 comma 1 right parenthesisC(59,1)

Part 11

​Thus, the total number of outcomes containing any six​ numbers,

55

of which are from the six numbers drawn on the ticket and the rest are​ not, can be found using the multiplication principle and is equal to

Upper C left parenthesis 6 comma 5 right parenthesisC(6,5)times•Upper C left parenthesis 59 comma 1 right parenthesisC(59,1).

Find​ P(E). Round to nine decimal places and then convert the number to scientific notation.

​P(E)

equals=StartFraction n left parenthesis Upper E right parenthesis Over n left parenthesis s right parenthesis EndFractionn(E)n(s)

equals=StartFraction Upper C left parenthesis 6 comma 5 right parenthesis times Upper C left parenthesis 59 comma 1 right parenthesis Over 82 comma 598 comma 880 EndFractionC(6,5)•C(59,1)82,598,880

equals=4.286 times 10 Superscript negative 64.286×10−6

Part 12

​Thus, the probability of picking exactly

55

of the 6 numbers drawn is

4.286 times 10 Superscript negative 64.286×10−6.