Explanation Required ... please anyone who can help me?

Question

7. Some guy named Jim has come up with a scheme for eliminating coins. This scheme involves probability and the fact that most cash registers today are computers. Suppose that every cash register could be programmed with a random number generator; that is, suppose that it were possible to pick a random number from 1 to 99. Jim's system works as follows. Suppose you purchase items totaling $15.89. The computer would choose a number from 1 to 99 and then compare it with the cents portion of the purchase. In this case, if the random number is between 1 and 89 the price would be rounded up to $16; if it is between 90 and 99 the price would be rounded down to $15. For example, if you purchase a cup of coffee for $1.20, and the random number generator produces a random number from 1 to 20, the price is $2, but if it produces a number from 21 to 99 the price is $1. Use your problem solving skills to make several comments on this scheme.


10 points for best answer .... please help

Answer 1

Let's do a simple analysis of the expectation.
Let's suppose I purchase something for $X and Y cents. Without this scheme I'd pay exactly that.
Let's see if Jim's scheme charges the customer more or less than this on average.

If the random number gives 1 to Y, I pay X+1.
If it gives Y+1 to 99, I pay X.

P(Cost = X+1) = Y/99
P(Cost = X) = (99-Y)/99
E(Cost) = (X+1)*Y/99 + X*(99-Y)/99
= X + Y/99

Without Jim's scheme, I'd pay X + Y/100 < X + Y/99
So the customer pays more on average.
This scheme slightly benefits the store and overcharges the customer.

If the number generation was from 1 - 100 or 0 - 99, it would have worked out equally.
Come to think about it, 0-99 makes the most sense. How does Jim's scheme work if hte item cost $15.00?

Answer 2

The scheme is intended to come out with a net of zero on each transaction (on the average). This is so that the company isn't making/losing money and so the consumer doesn't feel cheated.

Let's start with $15.89. Here if we get a number between 1 and 89 (probability of 89/99) we would round up to $16. That's a gain of 11 cents for the company.

And if we get 90 through 99 (10/99) we would round down to $15. That's a loss of 89 cents for the company.

Notice that these two products are not equal. There should be a zero net sum but there isn't.
89/99 * 11 ≠ 10/99 * 89

So how would we correct it? We should instead choose a number from 1 to *100*. If the value is less than or equal to the cent amount round up as before, and if it is greater than the number round down.

This way the amounts work out as follows:
Rounding up 89% (89/100) of the time, company makes 11 cents.
Rounding down 11% (11/100) of the time, company returns 89 cents to the customer.

Notice that these two values are equal so there is no net gain/loss to either the company or the customer.
89/100 * 11 = 11/100 * 89

For completeness, let's just enumerate some of the possible change amounts:

0:
100% of the time amount goes down by 0.
0% of the time, amount goes up by 100.
100 x 0 = 0 x 100

1:
99% of the time amount goes down by 1:
1% of the time amount goes up by 99
99 x 1 = 1 x 99

etc.

50:
50% of the time amount goes down by 50:
50% of the time amount goes up by 50:
50 x 50 = 50 x 50

etc.

98:
2% of the time amount goes down by 98:
98% of the time amount goes up by 2:
2 x 98 = 98 x 2

99:
1% of the time amount goes down by 99:
99% of the time amount goes up by 1:
1 x 99 = 99 x 1

The revised method looks good to me... but can Jim convince the consumer? Perhaps... maybe they'll see it as a mini-lottery each time they play... I mean *pay*.

Answer 3

I wouldn't buy it. Coins are fine with me except for nickels. I hate those more than pennies.