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Figure This!

Quick Answer:

The probability is 1/6.



Complete Solution:

There are several ways to approach this problem.

The table shows all the possibilities if the checks are labeled a, b, and c, and the corresponding envelopes A, B, and C. There are six different possibilities, each one of them equally likely.

a b c
a c b
b a c
b c a
c a b




Only one, however, has all three checks in the matching envelopes. Since exactly 1 case out of 6 has the checks in the correct envelopes, the probability is 1/6.

Alternate Approach 1:
Draw a tree diagram. In the following diagram, aA indicates that check a is in envelope A. Like the table shown earlier, the tree diagram indicates that there is only 1 correct outcome out of 6 equally likely possibilities. Therefore, the probability of the correct placement is 1/6.

Alternate Approach 2:
You also can consider the problem geometrically. Draw a rectangle like the one below to represent all the possibilities in this situation. Because the chance that the first check is placed in the correct envelope is 1 in 3, shade 1/3 of the rectangle to indicate this outcome.

If the first check is placed correctly, there are only two possible envelopes for the second check. This means that the chance that the second check is placed in the correct envelope is 1 in 2. To indicate this outcome, darken 1/2 of the previously shaded area.

If the first two checks are placed correctly, then the final check also must be placed in the correct envelope. No further shading is necessary. The darker shading represents 1/6 of the entire rectangle, so the probability that all three checks are placed correctly is 1/6.

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