Consider a situation where there an election where there are 8 total candidates, 4 democrats and 4 republicans. First, in parallel, there is a democratic runoff and republican runoff. For the democratic primary, the top 3 democrats are selected. Then, in the second runoff, the top two democrats go to another runoff. Finally, the top vote-getter of that runoff wins the primary. On the republican side, first the top two republican vote-getters arc selected. Afterwards, the second runoff takes place to select the republican candidate. Then, the democrat and republicans standoff. (a) what is the total number of different ways that the republican and democrat primaries can occur? (h) A car dealer in La Jolla says that you will win a Tesla if you correctly guess the sequence of the first, and second democrat primaries correctly. Another car dealder in Del Mar says that you will win a Tesla if you correctly guess the sequence of first and second republican primaries correctly? Calculate the probability of winning a Tesla for either dealership. Which dealership should you go to if you want to get a free Tesla?
a. Consider, the number of ways in which the 3 candidates survived the 1st democratic run off is 4C3 ways. Now, in the second runoff, 2 candidates can survive in 3C2 ways. Now, the final democratic candidate can be chosen in 2C1 ways. SImilarly, the top-two republicans can be selected in 4C2 ways and the final candidate can be chosen in 2C1 ways. Therefore, the total number of different ways that the republican and democrat primaries can occur is
4C3 x 3C2 x 2C1 x 4C2 x 2C1 = 4 x 3 x 2 x 6 x 2 = 288 ways.
b. Probability of getting a Tesla from La Jolla is 1/4 x 1/3 = 1/12.
Probability of getting a Tesla from Del Mar is 1/6.
Therefore, probability of getting a Telsa from either of the dealer is 1/12 + 1/6 = 1/4.
One should go with Del Mar as the probabilty of getting a Telsa is higher with this dealer.
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