at the first tri city meeting where 8 people are from twon A 7 people are from town B and 5 people are from town C. If the council consists of 5 people find the probability that 3 people are from town A and 2 people are from town B. How do i do this
To find the probability that 3 people are from town A and 2 people are from town B in a council of 5 people, we can use combinations.
Step 1: Calculate the total number of ways to choose 5 people from the total available population.
The total available population is 8 + 7 + 5 = 20 people.
So, the total number of ways to choose 5 people from a population of 20 is given by the combination formula: C(20, 5) = 20! / (5! * (20-5)! ) = 15504.
Step 2: Calculate the number of ways to choose 3 people from town A.
The number of ways to choose 3 people from town A is given by C(8, 3) = 8! / (3! * (8-3)! ) = 56.
Step 3: Calculate the number of ways to choose 2 people from town B.
The number of ways to choose 2 people from town B is given by C(7, 2) = 7! / (2! * (7-2)! ) = 21.
Step 4: Multiply the number of ways to choose 3 people from town A by the number of ways to choose 2 people from town B to get the desired outcome.
56 * 21 = 1176.
Step 5: Calculate the probability by dividing the desired outcome by the total number of possibilities.
Probability = Desired Outcome / Total Number of Possibilities = 1176 / 15504 ≈ 0.07576.
Therefore, the probability that 3 people are from town A and 2 people are from town B in a council of 5 people is approximately 0.07576 or 7.576%.
To find the probability, we need to understand the total number of possible outcomes and the desired outcomes.
Step 1: Determine the total number of possible outcomes.
In this case, the total number of outcomes can be calculated by selecting 5 people from a total of 20 people attending the meeting. So, we can use a combination formula, given by:
nCr = n! / (r! * (n-r)!)
where n = total number of people and r = number of people to be selected.
The total number of outcomes, denoted as "nCr", can be calculated as:
Total number of outcomes = 20C5 = 20! / (5! * (20-5)!)
Step 2: Determine the desired outcomes.
We want to find the probability that 3 people are from town A and 2 people are from town B.
The number of ways to select 3 people from town A is given by 8C3 (= 8! / (3! * (8-3)!)).
The number of ways to select 2 people from town B is given by 7C2 (= 7! / (2! * (7-2)!)).
Hence, the desired outcomes = (number of ways to select 3 people from town A) * (number of ways to select 2 people from town B) = 8C3 * 7C2
Step 3: Calculate the probability.
Finally, the probability is given by the ratio of desired outcomes to total outcomes:
Probability = (desired outcomes) / (total outcomes) = (8C3 * 7C2) / 20C5
To find the numerical value, you can compute the combination values and then perform the division.
Note: Make sure to use a calculator or software capable of handling factorials and large numbers for accurate calculations.