Roulette An American roulette wheel has 38 slots with...

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 $P(B)$ and $P(E)$. (c) Describe the event ${ }^{\alpha} B$ and $E^{\prime \prime}$ in words. Then find $P(B$ and $E)$ (d) Explain why $P(B$ or $E) \neq P(B)+P(E) .$ Then use the general addition rule to compute $P(B$ or $\bar{E})$

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 $P(B)$ and $P(E)$.
(c) Describe the event ${ }^{\alpha} B$ and $E^{\prime \prime}$ in words. Then find $P(B$ and $E)$
(d) Explain why $P(B$ or $E) \neq P(B)+P(E) .$ Then use the general addition rule to compute $P(B$ or $\bar{E})$

Key Concepts

Sample Space and Two-Way Table

In probability, the sample space is the set of all possible outcomes of an experiment. The information from the sample space can be organized into a two-way table when dealing with two categorical variables. This table presents the number of outcomes or probability for each combination of events, enabling a clearer understanding of the structure of the experiment and the relationship between the events.

Event Definition and Probability Calculation

An event is a subset of the sample space that satisfies some condition, and its probability is defined as the ratio of the number of outcomes in the event to the total number of outcomes in the sample space. In finite probability spaces, such as those involving roulette wheels or dice, calculating the probability involves counting the favorable outcomes for an event and then dividing by the total number of outcomes.

Complements of Events

The complement of an event A, often denoted by A' or ¬A, consists of all outcomes in the sample space that are not included in A. This concept allows one to compute probabilities by using the relation P(A') = 1 - P(A). Understanding complementary events is essential, especially when it is easier to determine the probability of an event's complement rather than the event itself.

Intersection and Union of Events

The intersection of two events (denoted by A and B) consists of all outcomes that are common to both events. The union of two events includes outcomes that are in at least one of the events, which may lead to their probabilities overlapping. This requires careful treatment to avoid double counting when calculating probabilities related to these combined events.

General Addition Rule

When events are not mutually exclusive, the general addition rule states that the probability of the union of two events is found by adding their individual probabilities and then subtracting the probability of their intersection. This rule is crucial for correctly calculating the probability of either one event or another occurring when the events can occur simultaneously.

Transcript

00:01 I have a question about roulette, and roulette's got 38 slots, numbers 1 -236, a 0, and a double -zero.

00:09 There's a figure that's shown there.

00:11 This is question 51 of section 5 .2, where 18 are red of the numbered slots, 18 are red, 18 are black, and 2, the 0 and double -zero are green.

00:24 So obviously when the wheels spun, you have a metal ball, drops in the middle, right, and it'll fall in the place.

00:30 So we're going to define events b as a ball that lands on black and e that ball that lands on an even number slot.

00:40 They're going to treat zero and double zero as even numbers.

00:45 So in the first part we need to make a two -way table that displays the sample space in terms of events, b and e, so black and even.

00:54 And so i've gone ahead and done that.

00:56 That's going to be over here.

00:58 We'll see that here in a second.

01:00 And then we're going to have to do a couple things.

01:03 We need to find the probability of b.

01:06 Okay, so probability of being black and probably being even.

01:11 And then part c described the event, a specific event in terms of, you know, probability of b and e.

01:30 And then d explained, why the probability of b or e does not equal the probability of b plus the probability of e, and then use the general addition rule to compute the probability of b or e.

01:47 Lots going on there.

01:48 All right.

01:50 So let's do the first part.

01:53 This was the two -way table, right, that we need to put together.

01:58 Right.

01:59 So then the next part would be to find the probability of b.

02:04 So we're good for part b here we're going to find the probability of b and the probability of e.

02:15 Okay, so the probability of b is going to be equal to 18.

02:26 So here's the b's, right? so the total is 18 out of 38.

02:31 So the probability of being black is 18 of 38, which is 0...