Chapter 5.1, Problem 22P

Expand Your Knowledge: Odds Against Betting odds are usually stated against the event happening (against winning).

The adds against event W are the ratio $\frac{P(notW)}{P\left(W\right)}=\frac{P\left(W^c\right)}{P\left(W\right)}$.

In horse racing, the betting odds are based on the probability that the horse does not win.

(a) Show that if we are given the odds against an event W as a:b, the

probability of notW is $P\left(W^c\right)=\frac{a}{a+b}$. Hint: Solve the equation $\frac{a}{b}=\frac{P\left(W^c\right)}{1-P\left(W^c\right)}$ for $P\left(W^c\right)$.

(b) In a recent Kentucky Derby, the betting odds for the favorite horse. Point Given, were 9 to 5. Use these odds to compute the probability that Point Given would lose the race. What is the probability that Point Given would win the race?

(c) In the same race, the betting odds for the horse Monarchos were 6 to 1. Use these odds to estimate the probability that Monarchos would lose the race. What is the probability that Monarchos would win the race?

(d) Invisible Ink was a long shot, with belting odds of 30 to 1. Use these odds to estimate the probability that Invisible Ink would lose the race. What is the probability the horse would win the race? For further information on the Kentucky Derby, visit the web site of the Kentucky Derby.

To determine

To explain: Whether the probability of ‘not an event W’ is $P\left(W^c\right)=\frac{a}{a+b}$ or not when the odds in favor of the event ‘W’ are $a:b$.

Answer to Problem 22P

Solution: Yes, the probability of ‘not an event W’ is $P\left(W^c\right)=\frac{a}{a+b}$.

Explanation of Solution

Given: The odds in the favor of an event ‘W’ are $a:b$ or $\frac{a}{b}$.

Calculation: The formula to calculate the odds against event A is:

$\text{Odds against A}=\frac{P\left(A^C\right)}{P\left(A\right)}$

Here, the formula for the odds against W can be written as:

$\text{Odds against W}=\frac{P\left(W^C\right)}{P\left(W\right)}$

Now,

$\frac{a}{b}=\frac{P\left(W^C\right)}{P\left(W\right)}$

It is known that the complement of an event W is defined as:

$P\left(W^c\right)=1-P\left(W\right)$

On simplifying further,

$\begin{array}{l}P\left(W^c\right)=1-P\left(W\right)\\ P\left(w\right)=1-P\left(W^c\right)\end{array}$

Therefore,

$\frac{a}{b}=\frac{P\left(W^C\right)}{1-P\left(W^C\right)}$

Now, cross multiply and simplify the provided expression as:

$\begin{array}{l}a\left(1-P\left(W^C\right)\right)=b\left(P\left(W^C\right)\right)\\ a-aP\left(W^C\right)=bP\left(W^C\right)\\ a=bP\left(W^C\right)+aP\left(W^C\right)\\ P\left(W^C\right)=\frac{a}{a+b}\end{array}$

Interpretation: So, if the odds in the favor of an event ‘W’ are $\frac{a}{b}$, the probability of ‘not event W’ is obtained as $P\left(W^c\right)=\frac{a}{a+b}$.

To determine

The probability using the odds that Point Given will lose the race and the probability that Point Given would win the race.

Answer to Problem 22P

Solution: The required probabilities are 0.64 and 0.34, respectively.

Explanation of Solution

Given: The betting odds for Point Given are 9 to 5 and these odds are used to calculate the probability. So, the required values are a = 9 and b = 5.

Calculation: Consider the information provided in part (a), where $P\left(W^c\right)=\frac{a}{a+b}$, which is the same as for Point Given losing. So, the probability that Point Given does not win can be calculated using the information of part (a) as:

$\begin{array}{c}P\left(\text{Point Given loses}\right)=\frac{9}{9+5}\\ =\frac{9}{14}\\ =0.64\end{array}$

Thus, the probability that Point Given wins the race is:

$\begin{array}{c}P\left(\text{Point Given wins}\right)\text{=1-}P\left(\text{Point Given loses}\right)\\ =1-0.64\\ =0.36\end{array}$

To determine

The probability that Monarchos will lose the race and also of him winning the race.

Answer to Problem 22P

Solution: The required probabilities are 0.86 and 0.14, respectively.

Explanation of Solution

Given: The betting odds for Monarchos are 6 to 1. So, the required values are a = 6 and b = 1.

Calculation: Consider the information provided in part (a), where $P\left(W^c\right)=\frac{a}{a+b}$, which is same as for Monarchos losing.

So, the probability that Monarchos does not win can be calculated using the information of part (a) as:

$\begin{array}{c}P\left(\text{Monarchos loses}\right)=\frac{6}{6+1}\\ =\frac{6}{7}\\ =0.86\end{array}$

Thus, the probability that Monarchos wins the race is:

$\begin{array}{c}\text{P}\left(\text{Monarchos wins}\right)\text{=1-}P\left(\text{Monachos loses}\right)\\ =1-0.86\\ =0.14\end{array}$

To determine

The probabilities that Invisible Ink will lose the race and also of him winning the race.

Answer to Problem 22P

Solution: The required probabilities are 0.97 and 0.03, respectively.

Explanation of Solution

Given: The betting odds for Invisible Ink are 30 to 1. So, the required values are a = 30 and b = 1.

Calculation: Consider the information provided in part (a), where $P\left(W^c\right)=\frac{a}{a+b}$, which is the same as for Invisible Ink losing.

So, the probability that Invisible Ink does not win can be calculated using the information of part (a) as:

$\begin{array}{c}P\left(\text{Invisible Ink loses}\right)=\frac{30}{30+1}\\ =\frac{30}{31}\\ =0.97\end{array}$

Thus, the probability that Invisible Ink wins the race is:

$\begin{array}{c}\text{P}\left(\text{Invisible Ink wins}\right)\text{=1-}P\left(\text{Invisible Ink loses}\right)\\ =1-0.97\\ =0.03\end{array}$

Hence, the probabilities are 0.97 and 0.03 respectively.