Roulette An American roulette wheel has 38 slots with numbers $1$through$36,0,$and $00$, as shown in the figure. Of the numbered slots, $18$are red, $18$are black, and $2$—the $0$and $00$—are 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 $P\left(B\right)$and$P\left(E\right)$.

c. Describe the event “$B$and $E$” in words. Then find the probability of this event.

d. Explain why $P(BorE)\ne P\left(B\right)+P\left(E\right)$. Then use the general addition rule to compute $P(BorE)$.

We have to determine a two way table displaying the sample space of events $B$and $E$.

Sample space of event $B$= $2,35,4,33,6,31,8,29,10,13,24,15,22,17,20,11,26,28$

Sample space of event $E$= $0,2,4,16,6,18,8,12,10,00,36,24,34,22,32,20,30,26,28$

We have to determine the probability of event $B$and event $E$.

We fill use the following formula :-

$$\begin{gathered} P\left(B\right)=\frac{TotalnumberofoutcomesinthesamplespaceintheeventB}{Totalnumberofoutcomes} \\ P\left(E\right)=\frac{TotalnumberofoutcomesinthesamplespaceintheeventE}{Totalnumberofoutcomes} \end{gathered}$$

$$\begin{gathered} P\left(B\right)=\frac{18}{38}=0.4737 \\ P\left(E\right)=\frac{19}{38}=0.5 \end{gathered}$$

We have to determine the probability of the event “$B$ and $E$”

We will use the following formula :-

$P(B\cap E)=\frac{Totalnumberofoutcomesinthesamplespaceintheevent(B\cap E)}{Totalnumberofoutcomes}$

The ball lands in the black even-numbered slot in the event "$B$and $E$."

Even-numbered slots that are black = $2,4,6,8,10,24,22,20,26,28$

 $\hspace{1em}P(B\cap E)=\frac{10}{38}=0.2632$

We have to determine$P(BorE)$

The chance of a ball falling on the black slot or an even numbered slot is denoted by the letter $P(BorE)$.

$$\begin{gathered} Fromtheabovesubparts,P\left(B\right)=0.4737,P\left(E\right)=0.5,P(BandE)=0.2632. \\ \hspace{1em}\hspace{1em}P(BorE)=0.4737+0.5–0.2632 \\ \hspace{1em}\hspace{1em}P(BorE)=0.7105 \end{gathered}$$