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?

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

The amount of bet is $1.

The net profit is $4999.

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.

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

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

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

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

(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.

(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.

(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.