Exercise 7 Page 680

We play a game that involves spinning a wheel. We will begin by finding the outcomes in the sample space.

Since there are four colors on the wheel, we can use the following abbreviations for each color.

• $Y$ — yellow
• $G$ — green
• $R$ — red
• $B$ — blue

Our sample space should cover the outcomes of two spins of the wheel. The first letter shows the outcome of the first spin, and the second letter corresponds to the color from the second spin. In our exercise, we want to determine whether randomly spinning blue and then green on the wheel are independent events. Therefore, we have to consider two events.

• $A\text{:}$ spinning blue on the first spin
• $B\text{:}$ spinning green on the second spin

Let's recall the Probability of Independent Events Formula.

Probability of Independent Events

Two events A and B are independent events if and only if the probability that both events occur is the product of the probabilities of the events. $P(A\text{ and }B)=P(A)\cdot P(B)$

In order to find the probability of event $A,$ we will use the theoretical probability. We need to find the ratio of the number of favorable outcomes to the number of possible outcomes. We will look for the outcomes where blue was obtained first. We can see that there are $4$ favorable outcomes in which the spin is blue, and the total number of possible outcomes is $16.$ Therefore, we are able to calculate $P(A).$ Now, let's find the probability of event $B.$ Notice that both the number of favorable and possible outcomes remain the same as for event $A.$ This is because we have also $4$ outcomes in which we get green on the second spin. Last we can calculate the probability of event $A\text{ and }B,$ which represents the situation in which we spin blue and then green on the wheel. Using the list of all possible outcomes we know there is only $1$ favorable outcome, and the number of possible outcomes is still $16.$ In order to determine whether two events $A$ and $B$ are independent, we need to multiply $P(A)$ and $P(B)$ and check whether the identity below holds. If it does, they are independent. Let's substitute $\frac{1}{4}$ for $P(A)$ and $\frac{1}{4}$ for $P(B),$ then compare their product with the previously found $P(A\text{ and }B).$ Since the identity $P(A\text{ and }B)=P(A)\cdot P(B)$ holds true, spinning blue and then green on the wheel are independent events.