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 two lines that touch each other on the table as in $1-2-3-4-5-6$

b. Betting on three numbers in a line, as in $1-2-3$

c. Betting on one number

d. Betting on four numbers that touch each other to form a square, as in $10-11-13-14$

e. Betting on two numbers that touch each other on the table, as in $10-11\text{or}10-13$

f. Betting on $0-00-1-2-3$

g. Betting on $0-1-2;\text{or}0-00-2;\text{or}00-2-3$

Short answer

(a) the probability of winning by Betting on two lines that touch each other on the table as in $1-2-3-4-5-6$is $0.16$

(b) the probability of winning by betting on three numbers in a line, as in $1-2-3$is $0.08$

(c) the probability of winning by betting on one number is $0.03$

(d) the probability of winning by betting on four numbers that touch each other to form a square, as in $10-11-13-14$ is $0.11$

(e) the probability of winning by betting on two numbers that touch each other on the table, as in $10-11$or $10-13$ is $0.053$

(f) the probability of winning by betting on $0-00-1-2-3$is $0.132$

(g) the probability of winning by betting on $0-1-2;\text{or}0-00-2;\text{or}00-2-3$is $0.08$

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 (part a)

Total number $=38$

In two lines, there are $6$numbers

Thus, we need to find the probability of winning by betting on two lines that touch each other on the table.

Require probability is
localid="1648093011236" $$\begin{gathered} P\left(E\right)=\frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}} \\ P\left(two\text{lines}\right)=\frac{6}{38} \\ P\left(two\text{lines}\right)=0.16 \end{gathered}$$

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

Total number$=38$

We need to find the probability of winning by betting on three numbers in a line, as in $1-2-3$

Require probability is

role="math" localid="1648093085351" $$\begin{gathered} P\left(E\right)=\frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}} \\ P\left(\text{three numbers in a line}\right)=\frac{3}{38} \\ P\left(\text{three numbers in a line}\right)=0.08 \end{gathered}$$

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$=38$

We need to find the probability of winning betting on one number

Require probability

$$\begin{gathered} P\left(E\right)=\frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}} \\ P\left(\text{one number}\right)=\frac{1}{38} \\ P\left(\text{one number}\right)=0.03 \end{gathered}$$

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 number$=38$

We need to find the probability of winning betting on four numbers that touch each other to form a square, as in $10-11-13-14$

Require probability is

role="math" localid="1648093584894" $$\begin{gathered} P\left(E\right)=\frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}} \\ P(fournumbersthatformasquare)=\frac{4}{38} \\ P(fournumbersthatformasquare)=0.11 \end{gathered}$$

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$=38$

we need to find the probability of winning betting on two numbers that touch each other on the table. as in $10-11\text{or}10-13$

Require probability is

$$\begin{gathered} P\left(E\right)=\frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}} \\ P(twonumbersthattoucheachother)=\frac{2}{38} \\ P(twonumbersthattoucheachother)=0.053 \end{gathered}$$

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

Total number$=38$

$0-00-1-2-3$consists of $5$numbers

Thus, we need to find the probability of winning by betting on $0-00-1-2-3$

Require probability is

$$\begin{gathered} P\left(E\right)=\frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}} \\ P(0-00-1-2-3)=\frac{5}{38} \\ P0-00-1-2-3)=0.132 \end{gathered}$$

Given information (part g)

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 g)

Total number$=38$

We need to find the probability of wining by betting on $0-1-2$; or $0-00-2$; or $00-2-3$

Require probability is

$$\begin{gathered} P\left(E\right)=\frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}} \\ P(0-1-2;0-00-2;00-2-3)=\frac{3}{38} \\ P(0-1-2;0-00-2;00-2-3)=0.08 \end{gathered}$$