The composition of the 108 congress was 51 republican 48 democrats and 1 independent.a committee on aid to higher education is to be form with 3 senators to be chosen random to head the committee. find the probability that the group of 3 consist of, all republicans , all democrats, 1 republican, 1 democrat, 1 independent?
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
There is no replacement.
All Repub. = 51/108 * 50/107 * 49/106 = ?
All Dem = 48/108 * 47/107 * 46/106 = ?
One each = 1/108 * 51/107 * 48/106 = ?
To find the probabilities, we first need to determine the total number of possible outcomes.
The total number of ways to choose 3 out of 100 senators (excluding the independent senator) is given by the combination formula:
C(n, r) = n! / (r!(n-r)!)
where n is the total number of senators and r is the number of senators to be chosen.
For our case, n = 100 and r = 3.
So, the total number of possible outcomes is:
C(100, 3) = 100! / (3!(100-3)!) = 100! / (3!97!) = (100 * 99 * 98) / (3 * 2 * 1) = 161,700
Now, let's calculate the probability for each scenario:
1. All Republicans:
Out of the 51 Republican senators, we need to choose 3. So the number of favorable outcomes is:
C(51, 3) = 51! / (3!(51-3)!) = (51 * 50 * 49) / (3 * 2 * 1) = 22,825
The probability of choosing 3 Republicans is:
P(3 Republicans) = favorable outcomes / total outcomes = 22,825 / 161,700 = 0.1414 (or approximately 14.14%)
2. All Democrats:
Out of the 48 Democrat senators, we need to choose 3. So the number of favorable outcomes is:
C(48, 3) = 48! / (3!(48-3)!) = (48 * 47 * 46) / (3 * 2 * 1)= 17,296
The probability of choosing 3 Democrats is:
P(3 Democrats) = favorable outcomes / total outcomes = 17,296 / 161,700 = 0.1069 (or approximately 10.69%)
3. 1 Republican, 1 Democrat, 1 Independent:
We have 1 independent senator, 48 Democrat senators, and 51 Republican senators.
The number of favorable outcomes is: (Number of ways to choose 1 Independent senator) * (Number of ways to choose 1 Democrat senator) * (Number of ways to choose 1 Republican senator)
= C(1, 1) * C(48, 1) * C(51, 1)
= (1 * 48 * 51) = 2,448
The probability of choosing 1 Republican, 1 Democrat, and 1 Independent is:
P(1 Republican, 1 Democrat, 1 Independent) = favorable outcomes / total outcomes = 2,448 / 161,700 = 0.0151 (or approximately 1.51%)
Remember, the probabilities should add up to 1, so you can calculate the probability of the remaining case (1 Republican, 2 Democrats) by subtracting the probabilities of the above cases from 1.
Hope this helps! Let me know if you have any further questions.