Madison, Wisconsin has to select a representative from the city council to go to
ID: 3376194 • Letter: M
Question
Madison, Wisconsin has to select a representative from the city council to go to an economic development conference. It will also select an alternate representative, in case there is a conflict or emergency that prevents the representative from attending.
Madisons' policy says that these selections are to be made completely at random from the members of the city council. The council’s membership is made up of 7 Democrats and 5 Republicans. I'm interested in the political party affiliation (Democrat or Republican) of the two selected people.
Show all work-
1. Summarize these trials of selecting two people (representative and alternate representative) from the city council in a probability tree. Please include all the properties of the probability tree.
2. Construct a discrete probability distribution for the number of Democrats among the two selections. Make sure that you include all the properties of the discrete probability distribution.
3. From the discrete probability distribution, calculate the a.mean and b.standard deviation of the number of Democrats.
x= number of a particular outcomes P ? ?Explanation / Answer
Solution
Problem is to select 2 from amongst 12 city council members -7 Democrats and 5 Republicans. And, then work out the probability distribution of the X, E(X) and SD(X), where X = number of democrats in the selected two.
Back-up Theory
Number of ways of selecting r things out of n things is given by nCr = (n!)/{(r!)(n - r)!}..(1)
Probability of an event E, denoted by P(E) = n/N ………………………………………..(2)
where n = n(E) = Number of outcomes/cases/possibilities favourable to the event E and N = n(S) = Total number all possible outcomes/cases/possibilities.
Mean (average) of X = E(X) = sum{x.p(x)} summed over all possible values of x…..…. (3)
E(X2) = sum{(x2).p(x)} summed over all possible values of x…………………………...(4)
Variance of X = V(X) = E(X2) – { E(X)}2……………………………………………..(5)
Standard Deviation of X = SD(X) = sq.rt of V(X) …………………..………………..(6)
Preparatory Work
[vide (1)],
2 out of12 can be selected in 12C2 = (12!)/{(2!)(10)!} = 66. So, N of (2) is 66 …………(7)
Since only 2 are to b selected, X = 0, 1 or 2.
X = 0 => No Democrat => both are Republicans => number of selections
= 5C2 = (5!)/{(2!)(3)!} = 10…………………………………………………………….(8)
X = 1 => Exactly one Democrat => one Democrat and one Republicans => number of selections = (7C1)(5C1) = 35…………………………………………………………….(9)
X = 2 => Both are Democrat => number of selections = 7C2 = (7!)/{(2!)(5)!} = 21…….(10)
Probability Distribution ANSWER 1
x
0
1
2
Total
Probability p(x)
10/66 [(8)/(7)]
35/66 [(9)/(7)]
21/66 [(10)/(7)]
1
x.p(x)
0
35/66
42/66
77/66 = 7/6
x2.p(x)
0
35/66
84/66
119/66
Vide (3), Mean, µ = 7/6 ANSWER 2
Vide (4), Variance, ?2 = (119/66) - (7/6)2 = 25/132 = 0.1894
Vide (5), Standard Deviation, ? = sqrt(0.1894) = 0.44 ANSWER 3
DONE
x
0
1
2
Total
Probability p(x)
10/66 [(8)/(7)]
35/66 [(9)/(7)]
21/66 [(10)/(7)]
1
x.p(x)
0
35/66
42/66
77/66 = 7/6
x2.p(x)
0
35/66
84/66
119/66
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