4 different letters are dropped on the floor and then inserted into four differently addressed envelopes. Find the probability that exactly 2 letters are inserted into the proper envelope.
It doesn't take long to list these cases, using uppercase letters to show correct placement and lowercase letters to show incorrect placement:
ABdc
AdCb
AcbD
dBCa
cBaD
baCD
We see there are 6 ways for this event to happen, or 4c2, out of 4! total possible ways of placing the letters in envelopes.
4c2/4! = ______
Well, let's see if I can calculate this probability without getting my numbers all mixed up! So we have four letters and four envelopes, right? We need to find the probability that exactly 2 letters end up in the right envelopes.
Now, let's consider the possibilities. The first letter has a 1/4 chance of being inserted into the correct envelope. Then the second letter will have a 1/3 chance of going to the right envelope. However, we only want exactly 2 letters to be in the right envelopes, so that means the remaining two letters must be swapped, right?
So, we multiply the probabilities together: (1/4) * (1/3) * (2/2) * (1/1). The (2/2) is because the third letter can either go into the right envelope or be swapped with the last letter, giving us only one possibility.
If we simplify that, we get (1/4) * (1/3) * (1/1) = 1/12. Therefore, the probability of exactly 2 letters being inserted into the proper envelope is 1/12.
Remember, probabilities can be tricky, so don't be too hard on me if I made a joke instead of giving you the serious answer you were looking for!
To find the probability that exactly 2 letters are inserted into the proper envelope, we need to first determine the total number of possible outcomes and the number of favorable outcomes.
Total number of possible outcomes (denominator):
Since there are 4 letters dropped on the floor and 4 envelopes, there are a total of 4! = 4 x 3 x 2 x 1 = 24 possible ways to insert the letters into the envelopes.
Number of favorable outcomes (numerator):
We want exactly 2 letters to be inserted into the proper envelopes. Let's consider the cases:
Case 1: First two letters are inserted into the proper envelopes.
In this case, the first letter can be inserted into its corresponding envelope in 1 way, and the second letter can also be inserted into its envelope in 1 way. The remaining 2 letters have 2! = 2 x 1 = 2 ways to be inserted into the incorrect envelopes. Therefore, there are 1 x 1 x 2 = 2 favorable outcomes for this case.
Case 2: Last two letters are inserted into the proper envelopes.
Following the same logic as in case 1, we have 2 favorable outcomes for this case.
The total number of favorable outcomes is 2 + 2 = 4.
Therefore, the probability that exactly 2 letters are inserted into the proper envelope is:
P(exactly 2 letters inserted correctly) = Number of favorable outcomes / Total number of possible outcomes
= 4 / 24
= 1 / 6
≈ 0.1667
So, the probability is approximately 0.1667 or 16.67%.
To find the probability that exactly 2 letters are inserted into the proper envelope, we need to determine the total number of possible arrangements and the number of favorable arrangements.
Since there are 4 different letters and 4 differently addressed envelopes, there are 4! (4 factorial) total possible arrangements. The factorial, denoted by !, means multiplying a number by all of the positive integers less than it.
4! = 4 x 3 x 2 x 1 = 24
Now, let's determine the number of favorable arrangements where exactly 2 letters are inserted into the proper envelope.
Step 1: Choose the 2 letters that will be inserted correctly. This can be done in C(4, 2) ways, which represents choosing 2 items out of 4.
C(4, 2) = 4! / (2! x (4-2)!) = 6 ways
Step 2: Now, we need to determine the number of ways the remaining 2 letters can be inserted incorrectly. This can be done in C(2, 2) ways, which represents choosing 2 items out of 2.
C(2, 2) = 2! / (2! x (2-2)!) = 1 way
Step 3: Multiply the results from Step 1 and Step 2 to get the total number of favorable arrangements.
Total favorable arrangements = 6 x 1 = 6 ways
Finally, divide the number of favorable arrangements by the total number of possible arrangements to get the probability.
Probability = Number of favorable arrangements / Total possible arrangements
Probability = 6 / 24
Probability = 1/4
Therefore, the probability of exactly 2 letters being inserted into the proper envelope is 1/4 or 0.25.