34. Terry puts three different letters randomly in three envelopes with different addresses, Find the probability thatat least one letter is put in the envelope with correct address.
Question
- Terry puts three different letters randomly in three envelopes with different addresses, Find the probability thatat least one letter is put in the envelope with correct address.
Solution
To solve this problem, we will use the concept of probability and the principle of inclusion and exclusion.
Step 1: Identify the total number of outcomes Terry has 3 letters and 3 envelopes, so the total number of ways he can put the letters in the envelopes is 3!, which is 321 = 6.
Step 2: Identify the number of outcomes where no letter is in the correct envelope This is a derangement problem, where we want to find the number of permutations of the letters where no letter appears in its original position. The number of derangements of 3 items is 2.
Step 3: Calculate the probability that no letter is in the correct envelope The probability is the number of successful outcomes divided by the total number of outcomes, so the probability that no letter is in the correct envelope is 2/6 = 1/3.
Step 4: Calculate the probability that at least one letter is in the correct envelope The probability that at least one letter is in the correct envelope is 1 minus the probability that no letter is in the correct envelope. So, the probability that at least one letter is in the correct envelope is 1 - 1/3 = 2/3.
So, the probability that at least one letter is put in the envelope with the correct address is 2/3.
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