Inclass Worksheet Week 9 Break out Room 5 PDF

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Summary

In-class assignment done in groups; deals with expected value; MGF 1106....

Description

PART 1- EXPECTED VALUE A probability distribution chart is used to quickly show various events and their probabilities. The events are called random variables and are labeled x. At Tucson Raceway Park, your horse, Soon-to-be-Glue, has a probability of 1/20 of coming in first place, a probability of 1/10 of coming in second place, and a probability of 1/4 of coming in third place. First place pays $4,500 to the winner, second place $3,500 and third place $1,500. We want to know if it is worthwhile to enter the race if it costs $1,000 to enter. To do this, we need to know what our expected return is. Notice that in this problem, x is your winnings, or how much money you walk away with. a) If you win the first prize what is your net return? Remember that you need to pay $1000 to enter the race. $3,500 b) What is the net return if you win the second prize? $2,500 c) What is your net return for the third prize? $500 d) What if you do not win a prize? What is your net return then? You lose $1,000. Net return= $-1,000 e) What is the probability of not winning? Remember that the total probability is always 1. 12/20 = 3/5 f) Use all of the above information to complete the probability distribution table below. Remember that x is your winnings, not the advertised prize. 1st Place

2nd Place

3rd Place

Losing

x 3500

2500

500

-1000

P(x) 1/20

1/10

1/4

3/5

g) What is the expected net return? (3500x1/20)+2500x1/10+(500x1/4)+(-1000x3/5)= 175+250+125-600=-50

h) Do you think it is worthwhile to enter the race? - No because the expected value is negative.

--------------------------STOP HERE--------------------------

PART 2- EXPECTED VALUE Cont.

You decide to enter a contest where you pay $5 for a chance at the spinner. If you spin a 0, you will win $100 and if you spin an odd number, you will win $10. What are your expected winnings?

Spin 0

Spin odd number

Entry fee

x

$95

$5

$-5

P(x)

1/10

1/2

4/10

Expected Value is= 9.5+2.5+(-2)=$10

--------------------------STOP HERE-------------------------PART 3 - STANDARD DEVIATION In the playground, you come across the following group of kids: Bob, age 9, Frank, age 11, James, age 12, Greg, age 12, John, age 8 and Brian, age 14 1) What is the average/mean of their ages? -

The average/mean of their ages is 11.

Let’s say we wanted to see, on average, how far away each of their ages was from the mean. Let’s make a chart to keep track of the information. Name

Age

Bob

Mean of Ages

Age – Mean age

9 11

-2

Frank

11 11

0

James

12 11

1

Greg

12 11

1

John

8 11

-3

Brian

14 11

3

2) What happens when you now try to take the mean of the differences in the last column? When adding up the last column you get 0 then you divide it by 6 to get the mean, making the mean of the last column 0. Clearly, taking that mean is not a good way to measure the average distance. One thing we could do is square each of the distances so that each would then become positive. Let’s go back to our chart: Name Age Bob

Mean of Ages Age- Mean age

Square of the Difference

9 11

-2

4

Frank

11 11

0

0

James

12 11

1

1

Greg

12 11

1

1

John

8 11

-3

9

Brian

14 11

3

9

3) Now take the average of the squares. Though this time, instead of dividing by 6, divide by 5. (In a later statistics course they will explain why you divide by 1 less than the amount). 4.8

4) Does the number make sense as being the average distance from the mean? Explain why or why not. No, 4.8 is too big of a difference. 5) To reverse the’ squaring’ we did, take the square root of what you found in (3). 2.19 This is called the standard deviation of a set of numbers 6) What information does standard deviation give you regarding data? The standard deviation gives you an understanding of the variability within a data set. 7) List out the of the steps that you take to find the Standard Deviation: 1. Find the mean of the numbers 2. Subtract mean from each individual number 3. Square each deviation 4. Add them 5. Divide them by one less than the numbers in the set 6. Find square root

8) Create a chart and find the standard deviation of the following data: 27, 15, 43, 12, 52, 9 X

Mean

x-mean

(x-mean)^2

27

26.3

-0.7

0.49

15

26.3

-11.3

127.69

43

26.3

16.7

278.89

12

26.3

-14.3

204.49

52

26.3

25.7

660.49

9

26.3

-17.3

299.29

Average of last column- 314.268 | SqRt of 314.268=Standard Deviation= 17.72761...