Question

Use the following information to answer the next three exercises. The casino game, roulette, allows the gambler to bet on the probability of a ball, which spins in the roulette wheel, landing on a particular color, number, or range of numbers. The table used to place bets contains of $38$numbers, and each number is assigned to a color and a range.

Compute the probability of winning the following types of bets:

a. Betting on a color

b. Betting on one of the dozen groups

c. Betting on the range of numbers from $1\text{to}18$

d. Betting on the range of numbers$19-36$

e. Betting on one of the columns

f. Betting on an even or odd number (excluding zero)

Short answer

(a) The probability of winning betting on color is $0.94$.

(b) The probability of winning betting on one of the dozen groups is $0.32$.

(c) The probability of winning betting o the range of number from $1-18\text{is}0.474$.

(d) The probability of winning betting on the range of numbers from $19-36\text{is}0.474$.

(e) The probability of winning betting on the columns is $0.32$.

(f) The probability of winning betting on an even or odd (exclusive zero) is $0.94$.

Step-by-step solution

Given information (part a)

The casino game, roulette, allows the gambler to bet on the probability of a ball, which spins in the roulette wheel, landing on a particular color, number, or range of numbers. The table used to place bets contains of $38$numbers, and each number is assigned to a color and a range

Explanation

Total number$=38$

Now in roulette, there are two colors - black and red.

In a game, there are $18$red color numbers and $18$black color numbers. Thus, we need o find the probability of winning by betting on color.

Require probability is

$$\begin{gathered} P\left(E\right)=\frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}} \\ P(redorblack)=\frac{36}{38} \\ P(redorblack)=0.94 \end{gathered}$$

Given information (part b)

The casino game, roulette, allows the gambler to bet on the probability of a ball, which spins in the roulette wheel, landing on a particular color, number, or range of numbers. The table used to place bets contains of $38$numbers, and each number is assigned to a color and a range

Explanation (part b)

Total number$=38$

Now in roulette, there are three groups of dozen

In a game, there are $12$numbers in one group of dozen

thus, we need to find the probability of winning by betting o one group of dozen is

$$\begin{gathered} P\left(E\right)=\frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}} \\ P(onedozen)=\frac{12}{38} \\ P(onedozen)=0.32 \end{gathered}$$

Given information (part c)

The casino game, roulette, allows the gambler to bet on the probability of a ball, which spins in the roulette wheel, landing on a particular color, number, or range of numbers. The table used to place bets contains of $38$numbers, and each number is assigned to a color and a range

Explanation (part c)

Total number$=38$

In a game, range of number from $1-18$consists of $18$numbers

Thus, we need to find the probability of winning betting n the range of numbers from $1-18$.

role="math" localid="1648097342071" $$\begin{gathered} P\left(E\right)=\frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}} \\ P(range1-18)=\frac{18}{38} \\ P(range1-18)=0.474 \end{gathered}$$

Given information (part d)

The casino game, roulette, allows the gambler to bet on the probability of a ball, which spins in the roulette wheel, landing on a particular color, number, or range of numbers. The table used to place bets contains of $38$numbers, and each number is assigned to a color and a range

Explanation (part d)

Total numbers

In a game, the range of numbers from $19-36$consists of $18$numbers

Thus, we need to find the probability of winning betting on the range of numbers from $19-36$.

Required probability is

$$\begin{gathered} P\left(E\right)=\frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}} \\ P(range19-36)=\frac{18}{38} \\ P(range1-18)=0.474 \end{gathered}$$

Given information (part e)

The casino game, roulette, allows the gambler to bet on the probability of a ball, which spins in the roulette wheel, landing on a particular color, number, or range of numbers. The table used to place bets contains of $38$numbers, and each number is assigned to a color and a range

Explanation (part e)

Total number $=38$

In a game, there are $12$numbers in one column

Thus, we need to find the probability of winning betting on the columns

Require probability is

$$\begin{gathered} P\left(E\right)=\frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}} \\ P\left(onecolumn\right)=\frac{12}{38} \\ P\left(onecolumn\right)=0.32 \end{gathered}$$

Given information (part f)

The casino game, roulette, allows the gambler to bet on the probability of a ball, which spins in the roulette wheel, landing on a particular color, number, or range of numbers. The table used to place bets contains of $38$numbers, and each number is assigned to a color and a range

Explanation (part f)

Total number $=38$

In a game, there are $18$even numbers and $18$odd numbers

Thus, we need to find the probability of winning by betting on an even or odd (excluding zero).

Required probability is

$$\begin{gathered} P\left(E\right)=\frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}} \\ P\left(evenorodd\right)=\frac{18}{38}+\frac{18}{38} \\ P\left(evenorodd\right)=\frac{36}{38} \\ P\left(evenorodd\right)=0.94 \end{gathered}$$