In a certain election, the incumbent Republican will run against the Democratic nominee. There are three Democratic candidates, D1, D2 and D3, whose chances of gaining the Democratic nomination are .50, .35 and .15, respectively. Here are the chances that the Republican will win against each of these possible Democratic nominees:

vs. D1: 0.60 vs. D2: 0.50 vs. D3: 0.40

(a) Name (but do not give) the probability formula that is needed to find the chance that the Republican will win the election.

(b) Find the probability that the Republican will win the election.
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Im not sure how to start.

a

two rules -addition rule (special case for mutually exclusive events) and multiplication rule (for independent events)
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b
probability of winning against D1 = probability of D1 being nominee * probability of Beating D1 if nominee
= .5 *.6 = .3
same multiply for candidate D2
= .35 * .5 = .175
same for D3
.15 * .4 = .06
now chances of winning = sum - any intersections but there are no intersections, only one democratic candidate in the end so = .3+.175+.06 = .535

To find the probability that the Republican will win the election, you need to consider the probabilities of each possible scenario and calculate the overall probability using the formula for the law of total probability.

(a) The probability formula needed to find the chance that the Republican will win the election is the law of total probability. This formula states:

P(A) = P(A | B1) × P(B1) + P(A | B2) × P(B2) + ... + P(A | Bn) × P(Bn),

where P(A) represents the probability of event A occurring, P(A | Bi) represents the probability of event A occurring given event Bi has occurred, and P(Bi) represents the probability of event Bi occurring.

(b) To calculate the probability that the Republican will win the election, you need to consider the probabilities of each scenario and multiply them by the chances of each scenario occurring.

Let's calculate the probability using the law of total probability:

P(Republican wins election) = P(Republican wins election | D1) × P(D1) + P(Republican wins election | D2) × P(D2) + P(Republican wins election | D3) × P(D3)

Given the probabilities provided in the question:

P(Republican wins election | D1) = 0.60
P(Republican wins election | D2) = 0.50
P(Republican wins election | D3) = 0.40

And the chances of each Democratic nominee gaining the nomination:

P(D1) = 0.50
P(D2) = 0.35
P(D3) = 0.15

Now, substitute these values into the formula:

P(Republican wins election) = (0.60 × 0.50) + (0.50 × 0.35) + (0.40 × 0.15)

Calculate the values:

P(Republican wins election) = 0.30 + 0.175 + 0.06

Summing up the values:

P(Republican wins election) = 0.535

Therefore, the probability that the Republican will win the election is 0.535, or 53.5%.