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

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.

Quote:

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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?

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If we have N different items and N corresponding correct places, if the items are randomly assigned to the places there are 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.