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Roulette, An American roulette wheel has$38$slots with numbers $1$through $36,0,and00,$as shown in the figure. Of the numbered slots, $18$are red, $18$are black, and $2$鈥攖丑别 $0$and $00$鈥攁re green. When the wheel is spun, a metal ball is dropped onto the middle of the wheel. If the wheel is balanced, the ball

is equally likely to settle in any of the numbered slots. Imagine spinning a fair wheel once. Define events B: ball lands in a black slot, and E: ball lands in an even numbered slot. (Treat $0$and $00$as even numbers.)

(a) Make a two-way table that displays the sample space in terms of events B and E.

(b) Find P(B) and P(E).

(c) Describe the event 鈥淏 and E鈥� in words. Then find P(B and E). Show your work.

(d) Explain why P(B or E) 鈮� P(B) + P(E). Then use the general addition rule to compute P(B or E).

Step by step solution

01

Part (a) Step 1. Given Information 

$n=52$is the number of cards in a typical deck.

Number of red cards: $n_r=26$

Number of black cards in the deck: $n_b=26$

Number of black cards in the deck: $n_j=26$

Events:

Getting a jack is represented by the letter $J$

$R$stands for receiving a red card.

02

Part (a) Step 2. Concept Used   

An event is a subset of an experiment's total number of outcomes. The ratio of the number of elements in an event to the number of total outcomes is the probability of that occurrence and the use of the complimentary rule.

03

Part (a)  Step 3. Explanation  

The circumstances: black and even.

The two-way table is as follows:

04

Part (b) Step 1. Calculation  

$Probability=\frac{Numberoffavorableoutcomes}{Totalpossibleoutcomes}$

$$\begin{gathered} P\left(Black\right)=P\left(B\right) \\ \frac{18}{38}=0.4737 \\ P\left(Even\right)=P\left(E\right) \\ \frac{20}{38}=0.5263 \end{gathered}$$

05

Part (c) Step 1. Calculation  

$$\begin{gathered} Probability=\frac{Numberoffavorableoutcomes}{Totalpossibleoutcomes} \\ P(BandE)=\frac{10}{38}=0.2631 \end{gathered}$$

06

Part (d) Step 1. Calculation  

$$\begin{gathered} P\left(B\right)=0.4737 \\ P\left(E\right)=0.5263 \\ P(BandE)=0.2631 \\ Probability=\frac{Numberoffavorableoutcomes}{Totalpossibleoutcomes} \\ P(BorE)=P\left(B\right)+P\left(E\right)鈭�P(BandE) \end{gathered}$$

B and E do not have to be mutually exclusive.

$P(BorE)鈮�P\left(B\right)+P\left(E\right)$

That is, the letters B and E are found together.

Therefore,

$P(BorE)=\frac{18}{38}+\frac{20}{38}-\frac{10}{38}=\frac{28}{38}=0.7368$

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