Chapter 4.1, Problem 22P

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

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

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

1. (a) Show that if we are given the odds against an event W as a:b, the probability of not W is $P(W^c)=\frac{a}{a+b}.$ Hint: Solve the equation $\frac{a}{b}=\frac{P(W^c)}{1P(W^c)}$ for P(W^c).
2. (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?
3. (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?
4. (d) Invisible Ink was a long shot, with betting 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

Prove that the probability of not W is $P\left(W^c\right)=\frac{a}{a+b}$.

Explanation of Solution

The odds against an event W is defined as follows:

$\begin{array}{c}\text{Odds against}=\frac{P\left(notW\right)}{P\left(W\right)}\\ =\frac{P\left(W^c\right)}{P\left(W\right)}\end{array}$

Where,

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

The odds against event W is denoted as $a:b$.

That is,

$\begin{array}{c}\text{Odds against}=\frac{a}{b}\\ \frac{P\left(W^c\right)}{P\left(W\right)}=\frac{a}{b}\left\{\therefore \text{by the definition}\right\}\\ \frac{P\left(W^c\right)}{1-P\left(W^c\right)}=\frac{a}{b}\\ a\left[1-P\left(W^c\right)\right]=bP\left(W^c\right)\\ a-aP\left(W^c\right)=bP\left(W^c\right)\\ a=P\left(W^c\right)\left[a+b\right]\\ P\left(W^c\right)=\frac{a}{a+b}\end{array}$

Therefore, it is proved that the probability of not W is $P\left(W^c\right)=\frac{a}{a+b}$.

To determine

Find the probability that the horse will lose the race.

Find the probability that the horse will win the race.

Answer to Problem 22P

The probability that the horse will lose the race is 0.643.

The probability that the horse will win the race is 0.357.

Explanation of Solution

From Part (a), probability of not W is $P\left(W^c\right)=\frac{a}{a+b}$.

It is given that the points given for the betting odds for the favorite horse are 9 to 5.

That is, $a:b=9:5$

Let $W^c$ be the event that the horse will lose the race and W be the event that the horse will win the race.

The probability that the horse will lose the race is as follows:

$\begin{array}{c}P\left(W^c\right)=\frac{a}{a+b}\\ =\frac{9}{9+5}\\ =\frac{9}{14}\\ \approx 0.643\end{array}$

Thus, the probability that the horse will lose the race is 0.643.

The probability that the horse will win the race is as follows:

$\begin{array}{c}P\left(W\right)=1-P\left(W^c\right)\\ =1-0.643\\ =0.357\end{array}$

Thus, the probability that the horse will win the race is 0.357.

To determine

Find the probability that Monarchos will lose the race.

Find the probability that Monarchos will win the race.

Answer to Problem 22P

The probability that Monarchos will lose the race is 0.875.

The probability that Monarchos will win the race is 0.125.

Explanation of Solution

From Part (a), the probability of odds against is $P\left(W^c\right)=\frac{a}{a+b}$.

The betting odds for the horse Monarchos are 6 to 1. That is, $a:b=6:1$.

Let $W^c$ be the event that the horse Monarchos will lose the race and W be the event that the horse Monarchos will win the race.

The probability that Monarchos will lose the race is as follows:

$\begin{array}{c}P\left(W^c\right)=\frac{a}{a+b}\\ =\frac{6}{6+1}\\ =\frac{6}{7}\\ =0.875\end{array}$

Thus, the probability that Monarchos will lose the race is 0.875.

The probability that Monarchos will win the race is as follows:

$\begin{array}{c}P\left(W\right)=1-P\left(W^c\right)\\ =1-0.875\\ =0.125\end{array}$

Therefore, the probability that Monarchos will win the race is 0.125.

To determine

Find the probability that Invisible Ink will lose the race.

Find the probability that Invisible Ink will win the race.

Answer to Problem 22P

The probability that Invisible Ink will lose the race is 0.968.

The probability that Invisible Ink will win the race is 0.032.

Explanation of Solution

The betting odds for the horse Invisible Ink are 30 to 1. That is, $a:b=30:1$.

Let $W^c$ be the event that the horse Invisible Ink will lose the race and W be the event that the horse Invisible Ink will win the race.

The probability that Invisible Ink will lose the race is as follows:

$\begin{array}{c}P\left(W^c\right)=\frac{a}{a+b}\\ =\frac{30}{30+1}\\ =\frac{30}{31}\\ \approx 0.968\end{array}$

Thus, the probability that Invisible Ink will lose the race is 0.968.

The probability that Invisible Ink will win the race is as follows:

$\begin{array}{c}P\left(W\right)=1-P\left(W^c\right)\\ =1-0.968\\ =0.032\end{array}$

Therefore, the probability that Invisible Ink will win the race is 0.032.