The organizer of a television show must select five people to participate in the show. The participants will be selected from a list of 24 people who have written in to the show. If the participants are selected randomly, what is the probability that the 5 youngest people will be selected?
There are 24C5 = 42,504 ways to pick 5 people.
Only one of those ways includes the 5 youngest.
To find the probability that the 5 youngest people will be selected, we need to determine the total number of possible outcomes and the number of favorable outcomes.
First, let's determine the total number of possible outcomes. There are 24 people in total, and we want to select a group of 5 out of these 24 people. This is a combination problem, and we can use the formula for combinations:
nCr = n! / (r!(n-r)!)
where n is the total number of items to choose from, and r is the number of items to choose.
In this case, we have 24 people to choose from, and we want to choose a group of 5, so:
24C5 = 24! / (5!(24-5)!) = 24! / (5!19!) = (24 * 23 * 22 * 21 * 20) / (5 * 4 * 3 * 2 * 1) = 53,130
So, there are 53,130 possible outcomes.
Now, let's determine the number of favorable outcomes. Since we want to select the 5 youngest people, we can assume that the position in which they are selected doesn't matter. So, we need to calculate the combination of selecting 5 out of the 5 youngest people from the total list of 24:
5C5 = 5! / (5!(5-5)!) = 1
Thus, there is only 1 favorable outcome.
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 = 1 / 53,130 = 0.000019 = 0.0019%
Therefore, the probability that the 5 youngest people will be selected is approximately 0.0019%.
To find the probability that the 5 youngest people will be selected, we need to calculate the total number of possible outcomes and the number of favorable outcomes.
Step 1: Calculate the total number of possible outcomes.
Since the participants are selected randomly, any combination of 5 people out of the 24 can be chosen. This can be calculated using the binomial coefficient, also known as "n choose k." In this case, n = 24 (total number of people) and k = 5 (number of people to select). The formula for the binomial coefficient is given by:
nCk = n! / (k! * (n-k)!)
Using this formula, we can calculate the total number of possible outcomes:
24C5 = 24! / (5! * (24-5)!)
Step 2: Calculate the number of favorable outcomes.
We are interested in selecting the 5 youngest people, which means we want to choose 5 specific people out of the 24. Since the order does not matter, this can be calculated using the concept of permutations. The formula for permutations is given by:
nPk = n! / (n-k)!
Using this formula, we can calculate the number of favorable outcomes:
5P5 = 5! / (5-5)!
Step 3: Calculate the probability.
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
Probability = (5P5) / (24C5)
Now, let's calculate the probability step-by-step:
Total number of possible outcomes = 24C5
= 24! / (5! * (24-5)!)
= 24! / (5! * 19!)
= (24 * 23 * 22 * 21 * 20) / (5 * 4 * 3 * 2 * 1)
= 2,024,760
Number of favorable outcomes = 5P5
= 5! / (5-5)!
= 5! / 0!
= 120
Probability = (5P5) / (24C5)
= 120 / 2,024,760
≈ 0.0000593
Therefore, the probability that the 5 youngest people will be selected is approximately 0.0000593.