help with probability question please!

Suppose you have 10 cards, 5 labeled "placebo" and 5 labeled "test." You

place them at random into 10 envelopes, 5 labeled "placebo" and 5 labeled "test."

What is the probability that exactly one envelope and corresponding card have the

same label?

Any help is helpful

Re: help with probability question please!

Hey amma0913.

You need to think about the number of combinations that are possible and note that there arele only 10 matches (each being a 1-1 match) for this problem.

Fix 1 envelope and you have 10 possibilities, the next has 9, then 8 then so on down to 1. This is 10! possibilities for a particular sequence of envelopes.

Now we have to factor the permutations of the envelopes and this also happens to be 10! using a similar argument.

Thus the total number of envelope/card combinations is 10!*10! and since we have 10 possible outcomes we have for each outcome a probability of 1/(10!*10!) for each of the ten outcomes.

Now you need to use probability laws to get P(Card1=Envelope1 OR Card2=Envelope2 ... OR ... OR Card10=Envelope10).

The above assumes that every possible way of putting in the 10 cards in 10 envelopes is taken into account.

Re: help with probability question please!

Quote:

Originally Posted by

**amma0913** Suppose you have 10 cards, 5 labeled "placebo" and 5 labeled "test." You place them at random into 10 envelopes, 5 labeled "placebo" and 5 labeled "test."

What is the probability that exactly one envelope and corresponding card have the same label?

If we have N different items and N corresponding correct places, if the items are randomly assigned to the places there are $\displaystyle N!\sum\limits_{k = 0}^N {\frac{{\left( { - 1} \right)^k }}{{k!}}} $ ways that none of the N items is in the correct place. These are known as derangements.

For this question we select one to be correct and the derangement nine others.