[Math] Letters and envelopes probability

Question

(a) If three letters are placed at random in three envelopes, what is the probability that exactly one letter will be placed in the correct envelope?

(b) If n letters are placed at random in n envelopes, what is the probability that exactly n−1 letters will be placed in the correct envelopes

For (a) I was thinking it will be $${3 \choose 1}\cdot(1)^{1/3}\cdot(2)^{2/3}$$

is that right? what would it be for a general n term then?

Answer 1

For (a), split it into disjoint events and then add up their probabilities:

• The probability that only the 1st letter is in the correct envelope is $\frac16$
• The probability that only the 2nd letter is in the correct envelope is $\frac16$
• The probability that only the 3rd letter is in the correct envelope is $\frac16$

The overall probability is therefore:

$$\frac16+\frac16+\frac16=\frac12$$

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For (b), the probability is obviously $0$, since if $n-1$ letters are in the correct envelope, then the remaining letter has "nowhere else to go" but the correct envelope too...