Question

Odds. In Exercises 41–44, answer the given questions that involve odds.

Finding Odds in Roulette A roulette wheel has 38 slots. One slot is 0, another is 00, and the others are numbered 1 through 36, respectively. You place a bet that the outcome is an odd number.

a. What is your probability of winning?

b. What are the actual odds against winning?

c. When you bet that the outcome is an odd number, the payoff odds are 1:1. How much profit do you make if you bet \(18 and win?

d. How much profit would you make on the \)18 bet if you could somehow convince the casino to change its payoff odds so that they are the same as the actual odds against winning? (Recommendation: Don’t actually try to convince any casino of this; their sense of humor is remarkably absent when it comes to things of this sort.)

Short answer

(a) The probability of winning is 0.474.

(b) The actual odds against winning are 10:9.

(c) If the amount of bet is $18, the profit made is $18.

(d) A profit of $20 is made on a bet of $18 if the payoff odds are equal to the actual odds against winning.

Step-by-step solution

Given information

A roulette wheel consists of 38 slots, numbered 0,00,1,2,3,…,36

A bet is won if the outcome is an odd number.

Define probability and odds

Probabilityis computed by dividing the number of outcomes in favor of event A by the total number of outcomes.

Mathematically, it is written as

$P\left(A\right)=\frac{\text{Number}\text{of}\text{outcomes}\text{in}\text{favor}\text{of}\text{A}}{\text{Total}\text{number}\text{of} \text{outcomes}}$.

Odds are computed as the ratio of two probabilities or values.

• The formula for actual odds in favor of an event A is

$\text{Actual}\text{odds}\text{in}\text{favor}=P\left(A\right):P\left(\overline{A}\right)$

• The formula for actual odds against event A is

$\text{Actual}\text{odds}\text{against}=P\left(\overline{A}\right):P\left(A\right)$

,where is the probability of the complementary event A or the event “not A”.

• The formula for the payoff odds is $\text{Payoff}\text{odds}=\left(\text{Net}\text{profit}\right):\left(\text{Amount}\text{of}\text{bet}\right)$.

Compute the probability to win

(a)

The total number of slots is 38.

The count of slots that have an odd number is 18.

Let $P\left(A\right)$ be the probability of winning the bet (getting an odd number).

Let $P\left(\overline{A}\right)$ be the probability of losing the bet (not getting an odd number).

These probabilities are calculated below:

$$\begin{gathered} P\left(A\right)=\frac{\text{Number}\text{of}\text{odd}\text{number}\text{slots}}{\text{Total}\text{number}\text{of}\text{slots}} \\ =\frac{18}{38} \\ =0.474 \end{gathered}$$

Therefore, the probability of winning is 0.474.

Compute the odds against winning

(b)

The probability of losing the bet is computed as follows.

$$\begin{gathered} P\left(\overline{A}\right)=1-P\left(A\right) \\ =1-\frac{18}{38} \\ =\frac{20}{38} \\ =0.526 \end{gathered}$$

Therefore, the probability of losing is 0.526.

The actual odds against winning are calculated as shown below:

$$\begin{gathered} \text{Actual}\text{odds}\text{against}\text{winning}=P\left(\overline{A}\right):P\left(A\right) \\ =\frac{\frac{20}{38}}{\frac{18}{38}} \\ =\frac{20}{18} \\ =10:9 \end{gathered}$$

Therefore, the actual odds against winning are 10:9.

Compute the profit from the payoff odds

(c)

The payoff odds are 1:1. That is, for a bet of $1, the profit earned is $1.

Mathematically,

$$\begin{gathered} \text{Payoff}\text{odds}=\left(\text{Net}\text{profit}\right):\left(\text{Amount}\text{of}\text{bet}\right) \\ =1:1 \end{gathered}$$

For a bet of $18, the net profit would be equal as the payoff odds are in the ratio of 1:1.

Therefore, the payoff odds are

$$\begin{gathered} \text{Payoff}\text{odds}=\left(\text{Net}\text{profit}\right):\left(\text{Amount}\text{of}\text{bet}\right) \\ =18:18 \end{gathered}$$

Therefore, the profit earned for a bet of $18 is $18.

Compute the profit from the payoff odds

d)

The actual odds against winning are 10:9.

The casino was convinced to change payoff odds equal to actual odds of winning.

The new payoff odds are 10:9.

$$\begin{gathered} \text{New}\text{payoff}\text{odds}=\left(\text{Net}\text{profit}\right):\left(\text{Amount}\text{of}\text{bet}\right) \\ =10:9 \end{gathered}$$

It means if the bet is of $9, then the profit earned is $10.

Let the bet be of $18. Then, according to the new payoff odds, the profit earned is

$$\begin{gathered} \text{New}\text{payoff}\text{odds}=\left(\text{Net}\text{profit}\right):\left(\text{Amount}\text{of}\text{bet}\right) \\ 10:9=\left(\text{Net}\text{profit}\right):\text{18} \\ \frac{10}{9}=\frac{\text{Net}\text{profit}}{18} \end{gathered}$$

$\text{Net}\text{profit}=20$

Thus, if the payoff odds are equal to the actual odds against winning, the profit earned on an $18 bet is $20.