If a permutation is chosen at random, what is the probability that it begins with at least 2 A's?
Fraction: 1/12
Decimal: 0.083333333
To find the probability that a randomly chosen permutation begins with at least 2 A's, we need to divide the number of permutations that satisfy this condition by the total number of possible permutations.
Let's consider the total number of possible permutations first. Assuming we have n distinct elements, the total number of permutations is given by n!.
To find the number of permutations that begin with at least 2 A's, we can count the number of possibilities. There are three cases to consider:
Case 1: The first two positions are both A's.
In this case, we have fixed the first two positions with A's. The remaining (n-2) elements can be arranged in (n-2)! ways.
Case 2: The first position is an A and the second position is a non-A.
In this case, we have fixed the first position with an A and the second position with a non-A. The remaining (n-2) elements can be arranged in (n-2)! ways.
Case 3: The first two positions are non-A's.
In this case, we have fixed the first two positions with non-A's. The remaining n elements can be arranged in (n-2)! ways.
Therefore, the number of permutations that begin with at least 2 A's is given by (n-2)! + (n-2)! + (n-2)! = 3!(n-2)! = 6(n-2)!.
Finally, the probability that a randomly chosen permutation begins with at least 2 A's is given by:
P = (number of permutations that begin with at least 2 A's) / (total number of permutations)
= (6(n-2)!)/(n!)
Note: The value of n should be known or given in order to obtain a specific numerical value for this probability.
To find the probability that a permutation begins with at least 2 A's, we need to determine two things: the total number of possible permutations and the number of permutations that meet the given condition.
Let's break it down step by step:
1. Determine the total number of possible permutations:
- If we have n distinct objects, the total number of permutations is n!.
2. Determine the number of permutations that begin with at least 2 A's:
- Suppose we have n total objects, including 2 A's and (n - 2) other distinct objects.
- We can choose the positions for the 2 A's in nP2 ways (permutation of 2 objects from n positions).
- The remaining (n - 2) objects can be arranged in (n - 2)! ways.
- Therefore, the number of permutations that begin with at least 2 A's is nP2 * (n - 2)!
3. Calculate the probability:
- The probability is the ratio of the number of permutations that meet the condition to the total number of permutations.
- Probability = (number of permutations that begin with at least 2 A's) / (total number of permutations)
To summarize, the probability that a permutation chosen at random begins with at least 2 A's is:
Probability = (nP2 * (n - 2)!) / n!