Question

Horse Racing. According to the Daily Racing Form, the probability is about$0.67$that the favorite in a horse race will finish in the money (first, second, or third place). In the next five races, what is the probability that the favorite finishes in the money
a. exactly twice?
b. exactly four times?
c. at least four times?
d. between two and four times, inclusive?
e. Determine the probability distribution of the random variable$X$, the number of times the favorite finishes in the money in the next five races.
f. Identify the probability distribution of$X$ as right skewed, symmetric, or left skewed without consulting its probability distribution or drawing its probability histogram.
g. Draw a probability histogram for $X$h. Use your answer from part (e) and Definitions $5.9$ and $5.10$ on pages $227$and $229$, respectively, to obtain the mean and standard deviation of the random variable $X$
i. Use Formulawidth="24" height="20" role="math">5.5on page $239$ to obtain the mean and standard deviation of the random variable$X$.

Short answer

The probability that the favorite finishes in the money in the next 5 races is that is

Step-by-step solution

Explanation

From the given information, According to the Daily Racing Form, the probability is about $0.67$that the favorite in a horse race will finish in the money (first, second, or third place) in the next 5 races.

Let $X$denotes the number of times the favorite finishes in the money in the next $5$races.

To solve parts $\left(a\right)-\left(d\right)$, we have to follow this procedure.

Step 1: Identify a success.

Success is when the favorite in a horse race will finish in the money.

Step 2: Determine $p$the success probability.

The probability that the favorite finishes in the money in the next 5 races is $67\%$that is $p=0.67$

Explanation

Step 3: Determine $n$, the number of trials.

The number of trials is the number of times the favorite finishes, which is five, so $n=5$

Step 4: The binomial probability formula for the number of successes, $X$, is

$P(X=x)={{(}^n}_x\left)p^x\right(1-p{)}^{n-x}$

Because $n=5$and $p=0.67$, the formula becomes

$P(X=x)=\left(\underset{x}{5}\right){(0.67)}^x{(0.33)}^{5-x}$

We see that $X$is a binomial random variable and has the binomial distribution with parameters $n=5$and $p=0.67$. Now we can solve parts (a)-(d) relatively easily.