Brandon Lang is a creative entrepreneur who has developed a novelty soap item called Jackpot to target consumers with a gambling habit. Inside each bar of Jackpot shower soap is a single rolled-up bill of U.S. currency. The currency (rolled up and sealed in shrink wrap) is appropriately inserted into the soap mixture prior to the cutting and stamping procedurel. The distribution of paper currency (per 1000 bars of soap) is given in the following table                                Distribution of Paper Currency                  Bill Denomination                        Number of Bills                              $1                                                520                              $5                                                260                             $10                                               120                             $20                                                 70                             $50                                                 29                             $100                                                                               Total                                           1000a.  What is the expected amount of money in a single bar of Jackpot soap?b. What is the standard deviation in a single bar of Jackpot soap?c. How many bars of soap would a customer have to buy so that, on average, he or she has purchased three bars containing $50 or $100 bill?d. If a customer buys 8 bars of soap, what is the probability that at least one of these bars contains a bill of $20 or larger?

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Asked Oct 24, 2019
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Brandon Lang is a creative entrepreneur who has developed a novelty soap item called Jackpot to target consumers with a gambling habit. Inside each bar of Jackpot shower soap is a single rolled-up bill of U.S. currency. The currency (rolled up and sealed in shrink wrap) is appropriately inserted into the soap mixture prior to the cutting and stamping procedurel. The distribution of paper currency (per 1000 bars of soap) is given in the following table

                                Distribution of Paper Currency

                  Bill Denomination                        Number of Bills

                              $1                                                520

                              $5                                                260

                             $10                                               120

                             $20                                                 70

                             $50                                                 29

                             $100                                                 

                              Total                                           1000

a.  What is the expected amount of money in a single bar of Jackpot soap?

b. What is the standard deviation in a single bar of Jackpot soap?

c. How many bars of soap would a customer have to buy so that, on average, he or she has purchased three bars containing $50 or $100 bill?

d. If a customer buys 8 bars of soap, what is the probability that at least one of these bars contains a bill of $20 or larger? 

 

    

 

 

                                                    

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Step 1

Note:

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Step 2

a. Computation of expected amount of money in a single bar of Jackpot soap:

The number of bills for $100 denomination is considered as 1 so that the total would be 1000.

The expected amount of money in a single bar of Jackpot soap is obtained as 5.97, from the calculation given below:

Number of Bills
P(x)
(per 1000 bars)
520
=0.52
1000
1
520
260
=0.26
260
5
1000
120
0.12
10
120
1000
70
0.07
70
1000
29
=0.029
1000
50
29
-0.001
100
1
1000
1000
1
E(x) -Σx.P(X = x)
($1x0.52) ($5x 0.26) + ($10x 0.12)+
($20 x 0.07)+($50x 0.029)+($100 x 0.001)
= 5.97
20
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Number of Bills P(x) (per 1000 bars) 520 =0.52 1000 1 520 260 =0.26 260 5 1000 120 0.12 10 120 1000 70 0.07 70 1000 29 =0.029 1000 50 29 -0.001 100 1 1000 1000 1 E(x) -Σx.P(X = x) ($1x0.52) ($5x 0.26) + ($10x 0.12)+ ($20 x 0.07)+($50x 0.029)+($100 x 0.001) = 5.97 20

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Step 3

b. Computation of standard deviation in a single bar of Jackpot soap:

The standard deviation in a single bar o...

V(1) = Σ Ε(x - Ε[;)}
(s1-5.97) (0.52)+($5-5.97) (0.26) +(S10-5.97) (0.12)+
|($20-5.97) (0.07)+ (S50-5.97) (0.029)+($100-5.97) (0.001)
E
=12.84 0.24+1.95+13.78+56.22 8.84
=93.88
S(x)=V(x)
=93.88
=9.69
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V(1) = Σ Ε(x - Ε[;)} (s1-5.97) (0.52)+($5-5.97) (0.26) +(S10-5.97) (0.12)+ |($20-5.97) (0.07)+ (S50-5.97) (0.029)+($100-5.97) (0.001) E =12.84 0.24+1.95+13.78+56.22 8.84 =93.88 S(x)=V(x) =93.88 =9.69

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