What is the probability that if we select four distinct integers uniformly at random from the set {1,2,…,20} and arrange them in ascending order, the third number is equal to 7?
To find the probability that the third number is equal to 7 when selecting four distinct integers uniformly at random from the set {1, 2, ..., 20} and arranging them in ascending order, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Let's break down the steps to approach this problem:
Step 1: Total number of outcomes
Since we are selecting four distinct numbers from a set of 20, the total number of outcomes is given by choosing 4 numbers out of 20, which can be represented as "20 choose 4" or written as (20 C 4). We can use the formula for combinations to calculate this value:
nCr = n! / (r!(n-r)!)
For this problem, n = 20 and r = 4, so we need to calculate (20 C 4).
(20 C 4) = 20! / (4!(20-4)!) = (20! / (4!16!))
Step 2: Number of favorable outcomes
To count the favorable outcomes, we need to calculate how many ways we can arrange the numbers so that the third number is equal to 7.
We fix the number 7 as the third number, so we have 3 more numbers to select from the remaining 19 numbers (excluding 7). The remaining 3 numbers can be chosen from the remaining 19 numbers, giving us (19 C 3) possible ways to choose these numbers.
To find the total number of arrangements with 7 as the third number, we multiply this by the number of possible arrangements of the remaining 3 numbers, which is simply 3!.
Therefore, the number of favorable outcomes is (19 C 3) * 3!.
Step 3: Calculating the probability
Now that we know the total number of outcomes and the number of favorable outcomes, we can calculate the probability.
Probability = Number of favorable outcomes / Total number of outcomes
Probability = [(19 C 3) * 3!] / (20 C 4)
Now, you can use a calculator or programming language to evaluate this expression and find the probability.
To find the probability that the third number is equal to 7 when selecting four distinct integers uniformly at random from the set {1, 2, ..., 20}, we need to determine the total number of possible outcomes and the number of favorable outcomes.
First, let's consider the total number of possible outcomes. We are selecting four distinct integers from a set of 20, so the total number of ways to choose them is given by the binomial coefficient "20 choose 4," denoted as C(20, 4) or
20! / (4!(20 - 4)!).
C(20, 4) = 20! / (4!(20 - 4)!)
= 20! / (4!16!)
= (20 × 19 × 18 × 17) / (4 × 3 × 2 × 1)
= 4845
Therefore, there are 4845 possible outcomes when selecting four distinct integers from the set {1, 2, ..., 20}.
Next, let's determine the number of favorable outcomes where the third number is equal to 7. Since the four integers must be arranged in ascending order, we can determine this by considering the two integers that come before and after 7.
The two integers that come before 7 must be chosen from the set {1, 2, ..., 6} (since they need to be smaller than 7), and there are 6 options. Similarly, the integer that comes after 7 must be chosen from the set {8, 9, ..., 20} (since it needs to be greater than 7), and there are 13 options.
Therefore, the number of favorable outcomes is given by 6 × 13 = 78.
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 78 / 4845
≈ 0.0161
So, the probability that the third number is equal to 7 is approximately 0.0161 or 1.61%.