1. In the game of “craps” two dice are thrown and the outcome of a bet is based

Question

1. In the game of “craps” two dice are thrown and the outcome of a bet is based on the sum of the two dice. If you bet $1 that the sum is “seven” then you win $4 or lose your dollar. The probability that you win is 6/36=1/6, and P(loss) = 5/6. Find a rough range for a) 100 plays, (b) 10, 000 plays. c) Compare 100 plays of this craps game with the class examples of 100 plays of bet $1 on red, 100 plays of bet $1 on “22” and 100 plays of Keno.   Which game would you play and why?

You must show your work when you compute the SD of the box!

[Hint: There are four steps in solving this problem. 1. You must first find the box model, the simplest model has six tickets in the box with some of the tickets +4 and others –1. You must determine how many of each of those two numbers are in the box. 2. Next find the Average of the box and the SD of the box, use “n” not “(n-1)” to compute the SD. 3. Third compute Expected(Winnings)=m·AveOfBox and SD(Winnings)=m·SDofBox, where m is the number of plays. 4. Finally the Rough Range is Expected(Win)±SD(Win).]

2. Work out the average and SD for the following list:

1, 3, 4, 5, 7

     Then work out the average and SD for the next list:

6, 8, 9, 10, 12

     Use n-1 in computing the SD.

     Are you surprised by the answers?

3. Use “n” in computing SD’s for this problem.

A list has 10 numbers, each number is a 1, or 2, or 3.

If the average is 2 and the SD is 0, find the list.

A second list has 10 numbers, each number is a 1, or 2, or 3.

If the SD is 1, find the list.

            c) Can the SD be bigger than 1?

            [This problem is solved by trial and error. Think what center and

            spread mean! You do not need to use every number for every list. If you  

            do not like the number 3, you may not have to use it]

4.   Find the population standard deviation for the following four populations:

a)     2, 3, 4, 5, 6

b)     2, 3, 4, 5, 6, 2, 3, 4, 5, 6

[Divide by 5 for the population in a), divide by 10 for the population in b).]

c)     2, -1, -1, -1

d)     2, -1, -1, -1, 2, -1, -1, -1

[Divide by 4 for the population in c), divide by 8 for the population in d).]

Explanation / Answer

Please post 1 question at a time:

4

a. 2,3,4,5,6

The mean is 4 ( as the average is (2+3+4+5+6)/5 = 4

The standard deviation is sqrt( (2-4)^2+...+(6-4)^2) / 5 = 1.414

b. Similarly, the mean for 2,3,4,5,6,2,3,4,5,6 is 4 ( please note that we need not use calculations to arrive at this answer)

This is repeat of part (a) , just that the numbers have eben repeated.

Hence, answer is 4

Even the standar deviation doesn't change. It will be 1.414

c. 2,-1,-1,-1

Here' the population mean is (2-1-1-1)/4 = -.25

The std deviation is sqrt( (2-.25)^2 +.. +(-1-.25)^2)/4 = 1.3

d. Same as case b, where the numbers are the same, the set of numbers have been repeated.

The population mean and the standard deviation will be same: Mean = -.25 and stdev = 1.3

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