Memory Match Problem #4

Probability Level 2

You are a playing a memory match game with your friends. You are using 50 matching pairs of cards (100 total). Unfortunately, you are losing.

To console yourself, you decide to figure out just how hard it is to get a match. To do this, you could find how many unmatching pairs of cards there are for every pair of matching cards.

So, what is the ratio of matching pairs to possible unmatching pairs?

Details and Assumptions:

This ratio is for the number of matching to unmatching pairs at the beginning of the game.
Order matters when considering possible pairs. For instance a pair of cards $(a, b)$ is considered different than a pair of cards $(b, a)$ . However, do not consider order for the matching pairs.

This problem is a part of the Memory Match Problem Series

\frac{1}{100}

\frac{1}{175}

\frac{1}{197}

\frac{1}{49}

1 solution

David Stiff
Jul 16, 2018

We have already been told the number of matching pairs, which is $50$ . To find the number of unmatching pairs, we can calculate the total number of possible pairs, and subtract from this the number of matching pairs.

The total number of possible pairs will be $100\cdot 99$ , or $9900$ (since we have 100 choices for the first card and 99 choices for the second card). Then we subtract the number of matching pairs, $50$ , from this to get $9850$ .

So the ratio of matching pairs to unmatching pairs is now $\dfrac{50}{9850}$ , which reduces to our final ratio of $\boxed{\dfrac{1}{197}}$ .

There may need to be some clarification. If the order in which you choose the cards matters then there are indeed a total of $9900$ possible pairs, but if order of choice does not matter then there are just $\dbinom{100}{2} = 4950$ possible pairs. This would result in $4900$ "unmatching pairs" and a desired ratio of $\dfrac{1}{98}$ . So you may want to consider mentioning that the order in which the cards are chosen matters.

Brian Charlesworth - 2 years, 10 months ago

Thanks Brian. That is unclear. I will update the problem to explain this.

David Stiff - 2 years, 10 months ago

If the order of the cards matters, then this applies to the matching pairs as well. There are 100 matching pairs (two ways to order each matching pair), so the answer is $\frac{100}{9900 - 100} = \frac{1}{98}.$

Jon Haussmann - 2 years, 10 months ago

Sorry if that was confusing Jon. The problem only stated that order matters when considering total possible pairs, not necessarily the matching pairs. I will note this in the problem.

David Stiff - 2 years, 10 months ago

Did same here.

D K - 2 years, 10 months ago

Memory Match Problem #4

1 solution

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