Question

In: Statistics and Probability

The City of Chicago has to select a representative from the city council to go to...

The City of Chicago 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. City 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.

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1. Summarize these trials of selecting two people (representative and alternate representative) from the city council in a 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 mean and standard deviation of the number of Democrats.

Expert Solution

1. Let R1 be the event that a Republican is selected and D1 be the event that a democrat is selected in the first round

Let us suppose that in the first round a representative is selected randomly out of 12 members.

The probability of choosein a Republican in the first round is

The probability of choosein a Democrat in the first round is

Let R2 be the event that a Republican is selected and D2 be the event that a democrat is selected in the second round

Depending on the result of the first round we have 2 more possibilities

A republican selected in the first round
- The probability of choosing a Republican in the second round given that republican was selected in the first round is

The probability of choosing a Democrat in the second round given that republican was selected in the first round is

A democrat selected in the first round
- The probability of choosing a Republican in the second round given that democrat was selected in the first round is

The probability of choosing a Democrat in the second round given that democrat was selected in the first round is

The probability tree is given below

2. Let D be the number of democrats selected

From the 4 leaf nodes of the tree we have 3 possible values of the number of democrats selected, which is 0,1 or 2

The probabilities are

The pmf of D is

3) The mean of number of democrats is the expected value of D, which is

For variance first we calculate the expecation of D2 as

Variance of D is

The standard deviation is

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