Question: Suppose that 4 thank-you notes are written and 4 envelopes are addressed. Accidentally, the notes…

Suppose that 4 thank-you notes are written and 4 envelopes are addressed. Accidentally, the notes are randomly inserted into the envelopes and mailed without checking the addresses. What is the probability that all the notes will be inserted into the correct envelopes? The probability is (Type an integer or decimal rounded to six decimal places as needed.)

Solution

To solve this problem, we need to calculate the probability that all 4 thank-you notes are inserted into the correct envelopes when they are randomly distributed. First, we determine the total number of ways to insert 4 notes into 4 envelopes. The total number of permutations of 4 notes is given by \(4!\). \[ 4! = 24 \] Next, we need to find the number of ways to correctly match all notes with their respective envelopes. There is only 1 way to do this, as each note must match with its specific envelope. Thus, the probability that all notes are correctly inserted is the ratio of the number of successful outcomes to the total number of possible outcomes. \[ \text{Probability} = \frac{1}{24} \] Finally, we convert this fraction to a decimal, rounded to six decimal places. \[ \text{Probability} = 0.041667 \] Therefore, the probability that all notes will be inserted into the correct envelopes is 0.041667.