Question

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

Kentucky Pick 4 In the Kentucky Pick 4 lottery, you can place a “straight” bet of \(1 by selecting the exact order of four digits between 0 and 9 inclusive (with repetition allowed), so the probability of winning is 1/10,000. If the same four numbers are drawn in the same order, you collect \)5000, so your net profit is $4999.

a. Find the actual odds against winning.

b. Find the payoff odds.

c. The website www.kylottery.com indicates odds of 1:10,000 for this bet. Is that description accurate?

Short answer

(a) The actual odds against winning are 9999:1.

(b) The payoff odds are 4999:1.

(c) No, the given odds of 1:10000 are not accurate as the actual odds in favor of winning are 1:9999.

Step-by-step solution

Given information

The probability of winning a lottery is $\frac{1}{10,000}$.

The amount of bet is $1.

The net profit is $4999.

Define probability and odds

Probabilityis computed by dividing the number of outcomes that lead to event A by the total number of outcomes.

Mathematically, the probability of an arbitrary event A is

$P\left(A\right)=\frac{\text{Number}\text{of}\text{outcomes}\text{that}\text{lead}\text{to}\text{A}}{\text{Total}\text{number}\text{of} \text{outcomes}}$

The odds of an event are written in the form of a ratio.

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

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

• The formula for the actual odds against event A is

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

Here, denotes the probability of the event “not A” or the complementary of A.

• The formula for payoff odds against event A is

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

Calculate the odds 

(a)

Let be the probability of winning the bet and be the probability of not winning the bet.

The probability of odds for winning is computed as follows.

$P\left(A\right)=\frac{1}{10000}$

$$\begin{gathered} P\left(\overline{A}\right)=1-\frac{1}{10000} \\ =\frac{9999}{10000} \end{gathered}$$

The actual odds against winning are computed as shown below:

$$\begin{gathered} \text{Actual}\text{odds}\text{against}\text{winning}=P\left(\overline{A}\right):P\left(A\right) \\ =\frac{\frac{9999}{10000}}{\frac{1}{10000}} \\ =\frac{9999}{1} \\ =9999:1 \end{gathered}$$

Therefore, the actual odds against winning are 9999:1.

Compute the payoff odds 

(b)

The net profit is $4999.

The amount of bet is $1.

The payoff odds are computed as shown below:

$$\begin{gathered} \text{Payoff}\text{odds}=\left(\text{Net}\text{profit}\right):\left(\text{Amount}\text{of}\text{bet}\right) \\ =4999:1 \end{gathered}$$

Therefore, the payoff odds are 4999:1.

Find if the description for odds is accurate 

(c)

The actual odds in favor of winning are calculated below:

$$\begin{gathered} \text{Actual}\text{odds}\text{in}\text{favor}\text{of}\text{winning}=P\left(A\right):P\left(\overline{A}\right) \\ =\frac{\frac{1}{10000}}{\frac{9999}{10000}} \\ =\frac{1}{9999} \\ =1:9999 \end{gathered}$$

Therefore, the actual odds in favor of winning are 1:9999.

The website displays the odds of 1:10000 for the bet.

Thus, it is inaccurate to say that the odds in favor of winning are 1:10000. The actual odds in favor of winning are 1:9999.