Question

In each of 12 races, the Democrats have a 60% chance of winning. Assume the races are independent of each other. Round your answers to three decimal places.a. What characteristics of this scenario indicate that you are working with Bernoulli trials? (3 points)b. What is the mean and standard deviation of the number of races won? (3 points)c. What is the probability that the Democrats will win the 7th race? (1 point)d. What is the probability that the Democrats will win all 12 races? (2 points)2

Answer 1

a)Since this scenario is about winning or losing an election, i.e., failure or success. Thus, this experiment has a Bernoulli distribution.

b)Also notice that since the elections are performed 12 times, the distribution of the whole experiment has a Binomial distribution. In this case, let p be the probability that the Democrats win. Then, we have the following:

`\begin{gathered} p=0.60 \\ 1-p=0.4 \\ n=12 \end{gathered}`

then, the mean and the standard deviation are:

`\begin{gathered} \mu=n\cdot p=12\cdot0.60=7.2 \\ \sigma=\sqrt[]{np(1-p)}=\sqrt[]{12\cdot0.60\cdot0.4}=\sqrt[]{2.88}=1.7 \end{gathered}`

c)To find the probability that the democrats will win the 7th race, we have to use the binomial probability function:

`P(X=x)=\binom{12}{x}(0.6)^x(0.4)^(12-x)`

in this case, x = 7, then we have:

`P(X=7)=\binom{12}{7}(0.6)^7(0.4)^(12-7)=792(0.6)^7(0.4)^5=0.23`

then, the probability that they win the 7th race is 23%

d) Finally, for the probability that the democrats win the 12 races, we can calculate with x = 12 to get:

`P(X=12)=\binom{12}{12}(0.6)^(12)(0.4)^(12-12)=(0.6)^(12)=0.002`

therefore, the probability that they win all the races is 0.2%