Question

1. when he was in first grade, harvey played games with spinners. one game he especially liked had two spinners and several markers that you moved around a board. you were only allowed to move if your color came up on both spinners. a. harvey always chose purple because that was his favorite color. what was the probability that harvey could move his marker? b. is the probability that harvey moves his marker calculated from a union or an intersection of events? c. was purple the best color choice? explain.

Explanation:

Step1: Determine probabilities for each spinner

The first spinner has 4 equal - sections, so the probability of getting purple on the first spinner, $P(P_1)=\frac{1}{4}$. The second spinner has 3 equal - sections, so the probability of getting purple on the second spinner, $P(P_2)=\frac{1}{3}$.

Step2: Calculate combined probability

Since the spins are independent events, the probability that both spinners land on purple is the product of their individual probabilities. Using the formula for independent events $P(A\cap B)=P(A)\times P(B)$, we have $P = P(P_1)\times P(P_2)=\frac{1}{4}\times\frac{1}{3}=\frac{1}{12}$.

Step3: Answer part b

Harvey can move only when purple comes up on both spinners. This is an intersection of events because we need the event of purple on the first spinner and the event of purple on the second spinner to occur simultaneously.

Step4: Answer part c

To determine if purple is the best choice, we need to compare the probabilities for other colors. For green on the first spinner and second spinner: $P(G_1)=\frac{1}{4},P(G_2)=\frac{1}{3},P(G_1\cap G_2)=\frac{1}{4}\times\frac{1}{3}=\frac{1}{12}$. For yellow on the first spinner and second spinner: $P(Y_1)=\frac{1}{2},P(Y_2)=\frac{1}{3},P(Y_1\cap Y_2)=\frac{1}{2}\times\frac{1}{3}=\frac{1}{6}$. Since $\frac{1}{6}>\frac{1}{12}$, purple is not the best color choice. Yellow has a higher probability of coming up on both spinners.

Answer:

a. $\frac{1}{12}$
b. Intersection of events
c. No, because the probability of yellow coming up on both spinners is $\frac{1}{6}$, which is higher than the probability of purple coming up on both spinners ($\frac{1}{12}$).