At the first tri-city meeting, there are 8 people from town A, 7 people from town B, and 5 people from town C. If a council consisting of 5 people is randomly selected, find the probability that 3 are from town A and 2 are from town B.
Without restrictions, number of councils is
C(20,5) or 15504
I will assume you want exactly 3 from A and 2 from B, so that number is C(8,3)xC(7,2) or 1176
so prob(of stated event) = 1176/15504
= 49/646 or .07585
Well, the total number of people attending the tri-city meeting is 8 + 7 + 5 = 20.
To find the probability of selecting a council with 3 people from town A and 2 people from town B, we need to consider both the number of ways to choose 3 people from town A (denoted as C(8, 3)) and the number of ways to choose 2 people from town B (denoted as C(7, 2)).
The total number of ways to choose 5 people from a group of 20 can be calculated as C(20, 5).
So, the probability can be calculated as:
( C(8, 3) * C(7, 2) ) / C(20, 5)
However, I must admit, my probability of understanding probability jokes is much higher than actually calculating them!
To find the probability, we need to determine the total number of ways to select 5 people out of the total 20 people.
The total number of ways to choose 5 people out of the 20 is given by the combination formula, denoted as nCr. It is calculated as:
nCr = n! / (r!(n-r)!)
Where n is the total number of people and r is the number of people chosen.
In this case, n = 20 and r = 5, so the combination formula becomes:
20C5 = 20! / (5!(20-5)!)
Next, we need to determine the number of ways to choose 3 people from town A out of the 8 available and 2 people from town B out of the 7 available.
The number of ways to choose 3 people from town A is given by the combination formula for 8C3:
8C3 = 8! / (3!(8-3)!)
Similarly, the number of ways to choose 2 people from town B is given by the combination formula for 7C2:
7C2 = 7! / (2!(7-2)!)
Finally, to find the probability, we need to divide the number of ways to choose 3 people from town A and 2 people from town B by the total number of ways to choose 5 people, which is 20C5:
Probability = (8C3 * 7C2) / 20C5
Now, we can calculate the probability step by step:
Step 1: Calculate 8C3
8C3 = (8! / (3!(8-3)!))
= (8! / (3!5!))
= (8 * 7 * 6 / (3 * 2 * 1))
= 56
Step 2: Calculate 7C2
7C2 = (7! / (2!(7-2)!))
= (7! / (2!5!))
= (7 * 6 / (2 * 1))
= 21
Step 3: Calculate 20C5
20C5 = (20! / (5!(20-5)!))
= (20! / (5!15!))
= (20 * 19 * 18 * 17 * 16 / (5 * 4 * 3 * 2 * 1))
= 15504
Step 4: Calculate the final probability
Probability = (8C3 * 7C2) / 20C5
= (56 * 21) / 15504
= 0.07653
Therefore, the probability that 3 people are from town A and 2 people are from town B is approximately 0.07653 or 7.653%.
To find the probability that 3 people from town A and 2 people from town B are selected in a council of 5 people, we need to calculate the total number of ways this can happen and divide it by the total number of possible councils.
First, let's determine the total number of ways to select 3 people from town A out of the 8 available. This can be done using the combination formula, which calculates the number of ways to choose a specific number of items from a larger set without regard to the order. The formula is:
nCr = n! / (r!(n-r)!)
where n is the total number of items and r is the number of items to choose.
In this case, we want to choose 3 people from town A out of 8, so we can calculate:
8C3 = 8! / (3!(8-3)!) = (8 * 7 * 6) / (3 * 2 * 1) = 56
Next, we need to determine the total number of ways to select 2 people from town B out of the 7 available:
7C2 = 7! / (2!(7-2)!) = (7 * 6) / (2 * 1) = 21
Now, we can calculate the total number of ways to select 3 people from town A and 2 people from town B. Since these two events are independent, we can use multiplication to calculate the total number of ways:
Total ways = 56 * 21 = 1,176
Finally, we need to calculate the total number of possible councils of 5 people, considering all the towns. This can be done using the combination formula once again:
Total councils = (8+7+5)C5 = 20C5 = 20! / (5!(20-5)!) = (20 * 19 * 18 * 17 * 16) / (5 * 4 * 3 * 2 * 1) = 15,504
Now we can calculate the probability by dividing the total number of ways to select 3 people from town A and 2 people from town B by the total number of possible councils:
Probability = Total ways / Total councils = 1,176 / 15,504 = 0.0758
Therefore, the probability that 3 people are from town A and 2 people are from town B in the randomly selected council of 5 people is approximately 0.0758 or 7.58%.