Clothes shopping!

Probability Level 4

You are shopping at the hottest new fashion outlet in town. While in the check-out line, you reach in your pocket and realize that in addition to your dollar bills, you have a quarter, dime, nickel, and two pennies. What is the probability that you will not be able to exactly match the change part of your bill with the 5 coins in your pocket?

Details and assumptions

Please write your answer in decimal form, rounded (if necessary) to the nearest hundredth.

A note on the source

This problem was taken from the Ole Miss Math Contest , which was hosted by Dr. David Rock of The University of Mississippi; the contest has not been so active recently and will probably be on hiatus for the foreseeable future. This problem was taken from the 'Algebra in Action' Challenge Mode section.

Image source: Wojtek Witkowski, via Unsplash archive


The answer is 0.76.

Dana Chiueh by dana chiueh , 21, Taiwan

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1 solution

Dana Chiueh
Sep 29, 2014

There are 100 change possibilities for the bill ranging from 00 to 99. A quarter, dime, nickel, and two pennies can account for the following possibilities:

00, 01, 02, 05, 06, 07, 10, 11, 12, 15, 16, 17, 25, 26, 27, 30, 31, 32, 35, 36, 37 40, 41, 42,

which represents 24 possibilities. Therefore, you can account for 24 100 \frac{24}{100} , but the question asks for the probability that you will not have the correct change or 76 100 \frac{76}{100} or 19 25 \frac{19}{25} = 0.76 \boxed{0.76} .

If anyone has any other way to solve this other than enumerating (counting), please do share the solution!

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Without enumerating, you can argue that we can account for 3 × 2 × 2 × 2 = 24 3 \times 2 \times 2 \times 2 = 24 possible cases, based on the number of coins of each type that we have.

We just have to be careful that there is no overlap, but that is somewhat easy to show.

Calvin Lin Staff - 6 years, 8 months ago

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Thank you! That is very helpful :)

dana chiueh - 6 years, 8 months ago

Strictly speaking, the problem asks which change parts we will not be able to match <with the coins in our pocket>, therefore it seems to me that 00 should be left out, since we can't match a .00 with the coins in our pocket.

Strangely enough, I was given a "right" answer with 0.77 (matching only 23 different possibilities), so I'm just adding my two cents (wink wink) here.

Carlos Merino - 5 years, 10 months ago

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