91Ó°ÊÓ

In a horse race, the probability of a favorite finish in the money is \(0.67\). Consider the next five races.

From the given information it can conclude that the number of times the favorite finish in the money follows a binomial distribution with \(n=5\) and \(p=0.67\).

Let the random variable \(X\) is that the number of times the favorite finish in the money among five races.

The probability of exactly two times the favorite finish in the money is

\(P(X=2)=\left ( \frac{5}{2} \right )0.67^{2}(1-0.67)^{5-2}\)

\(=0.1613\)

The probability of exactly four times the favorite finish in the money is

\(P(X=4)=\left ( \frac{5}{4} \right )0.67^{4}(1-0.67)^{5-4}\)

\(=0.3325\)

The probability of at least four times the favorite finish in the money is

\(P(X\geq 4)=P(X=4)+P(X=5)\)

\(=\binom{5}{4}0.67^{4}(1-0.67)^{5-4}+\binom{5}{5}0.67^{5}(1-0.67)^{5-5}\)

\(=0.3325+0.1350\)

\(=0.4675\)

The probability of between two and four times inclusively the favorite finish in the money is

\(P(2\leq X\leq 4)=P(X=2)+P(X=3)+P(X=4)\)

\(=\binom{5}{2}0.67^{2}(1-0.67)^{5-2}+\binom{5}{3}0.67^{3}(1-0.67)^{5-3}+\binom{5}{4}0.67^{4}(1-0.67)^{5-4}\)

\(=0.1613+0.3275+0.3325\)

\(=0.8213\)

Evaluate the probabilities at \(x=0,1,2,3,4,5\)

\(P(X=0)=\binom{5}{0}0.67^{0}(1-0.67)^{5-0}\)

\(=0.0039\)

\(P(X=1)=\binom{5}{1}0.67^{1}(1-0.67)^{5-1}\)

\(=0.0397\)

\(P(X=2)=\binom{5}{2}0.67^{2}(1-0.67)^{5-2}\)

\(=0.1613\)

\(P(X=3)=\binom{5}{3}0.67^{3}(1-0.67)^{5-3}\)

\(=0.3275\)

\(P(X=4)=\binom{5}{4}0.67^{4}(1-0.67)^{5-4}\)

\(=0.3325\)

\(P(X=5)=\binom{5}{5}0.67^{5}(1-0.67)^{5-5}\)

\(=0.1350\)

Combining this information in a table, it can obtain the probability distribution:

The binomial distribution is

• Symmetric if \(p=0.5\),
• Skewed to left if \(p>0.5\),
• Skewed to right if \(p<0.5\).

In this case, \(p=0.67>0.5\).

Therefore the distribution is skewed to left.

The height of the bars are equal to the probability and the bars are centered about the values of \(X\).

The mean

\(\mu =\binom{0\times 0.0039+1\times 0.0397+2\times 0.1613}{3\times 0.3275+4\times 0.3325+5\times 0.1350}\)

\(=3.3498\)

The standard deviation

\(\sigma =\sqrt{0^{2}\times0.0039+1^{2}\times0.0397+2^{2}\times0.1613+3^{2}\times0.3275+4^{2}\times0.3325+5^{2}0.1350-3.3498^{2}}\)

\(=1.0518\)

The mean

\(\mu =5\times0.67\)

\(=3.35\)

The standard deviation

\(\sigma =\sqrt{5\times0.67\times(1-0.67)}\)

\(=1.0514\)

Note that the small difference with the results of part (h) are due to rounding error.

Interpretation:

On average, the favorite finish in the money in \(3.35\) out of \(5\) horse races.