Question

use the spinner shown. it is equally probable that the pointer will land on any one of the regions. if the pointer lands on a borderline, spin again. if the pointer is spun twice, find the probability that it will land on a color other than blue for each spin. find the probability that the spinner will land on a color other than blue for each spin. (type an integer or a simplified fraction.)

Explanation:

Step1: Count total regions

The spinner has 8 equal regions (from the diagram: 2 blue, 2 red, 2 green, 2 yellow? Wait, no, looking at the spinner: blue regions: 2? Wait, let's count again. The spinner is divided into 8 parts? Wait, the colors: yellow, blue, red, green, red, blue, yellow, green? Wait, no, let's list the regions: yellow, blue, red, green, red, blue, yellow, green? Wait, no, maybe 8 equal sectors. Wait, blue regions: how many? Let's see the spinner: blue appears twice? Wait, the diagram: top blue, bottom blue, so 2 blue regions. Total regions: 8? Wait, let's check: yellow (2), blue (2), red (2), green (2). So total 8 regions.

Step2: Count non-blue regions

Non - blue regions: total regions - blue regions. Blue regions: 2. So non - blue regions: 8 - 2 = 6.

Step3: Calculate probability for one spin

Probability of landing on non - blue in one spin is number of non - blue regions divided by total regions. So $P(\text{non - blue})=\frac{6}{8}=\frac{3}{4}$.

Step4: Calculate probability for two independent spins

Since the spins are independent, the probability that both spins land on non - blue is $P = \frac{3}{4}\times\frac{3}{4}=\frac{9}{16}$? Wait, no, wait the question: "find the probability that it will land on a color other than blue for each spin". Wait, maybe I misread. Wait, the first part: "If the pointer is spun twice, find the probability that it will land on a color other than blue for each spin." Wait, no, the second part: "Find the probability that the spinner will land on a color other than blue for each spin." Wait, maybe the first part is a mis - read. Wait, the user's question: the problem is to find the probability that for each of the two spins, it lands on non - blue. Wait, but first, let's re - examine the spinner. Wait, maybe the spinner has 8 equal parts? Wait, let's count the regions again. Looking at the spinner: yellow, blue, red, green, red, blue, yellow, green. So 8 regions. Blue regions: 2 (top blue and bottom blue). So non - blue: 8 - 2 = 6. So probability for one spin: $\frac{6}{8}=\frac{3}{4}$. Then, for two spins, since the spins are independent, the probability that both are non - blue is $\frac{3}{4}\times\frac{3}{4}=\frac{9}{16}$? Wait, no, wait the question: "find the probability that it will land on a color other than blue for each spin". Wait, maybe the question is first, find the probability for one spin, and then for two spins? Wait, no, the problem says: "If the pointer is spun twice, find the probability that it will land on a color other than blue for each spin." Wait, maybe I made a mistake in counting the regions. Wait, let's look at the spinner again. The spinner: yellow, blue, red, green, red, blue, yellow, green. So 8 sectors. Blue: 2, so non - blue: 6. So probability for one spin: 6/8 = 3/4. Then, for two spins, the probability that each spin is non - blue is (3/4)*(3/4)=9/16. Wait, but the second part of the problem says: "Find the probability that the spinner will land on a color other than blue for each spin." Maybe that's a typo, and it's for two spins? Wait, no, let's check the original problem again.

Wait, the problem: "Use the spinner shown. It is equally probable that the pointer will land on any one of the regions. If the pointer lands on a borderline, spin again. If the pointer is spun twice, find the probability that it will land on a color other than blue for each spin.

Find the probability that the spinner will land on a color other than blue for each spin. (Type an integer or a simplified fraction.)"

Wait, maybe the first part is about two spins, but the second part is about one spin? Wait, no, the wording is confusing. Wait, let's re - analyze.

Wait, the spinner: let's count the number of blue regions. Looking at the diagram: blue appears twice (top and bottom). Total number of regions: let's count the number of colored sectors. Yellow: 2, blue: 2, red: 2, green: 2. So total 8 sectors.

Number of non - blue sectors: 8 - 2 = 6.

Probability of landing on non - blue in one spin: $P=\frac{\text{number of non - blue regions}}{\text{total number of regions}}=\frac{6}{8}=\frac{3}{4}$.

If we are spinning twice, and we want the probability that each spin (both spins) lands on non - blue, since the spins are independent events, we multiply the probabilities of each event. So $P(\text{non - blue on first spin and non - blue on second spin})=P(\text{non - blue}) \times P(\text{non - blue})=\frac{3}{4}\times\frac{3}{4}=\frac{9}{16}$.

Wait, but the second part of the problem says "Find the probability that the spinner will land on a color other than blue for each spin." Maybe it's a mistake, and it's asking for the probability for two spins. Let's confirm the number of regions again.

Alternative way: maybe the spinner has 8 equal parts. Blue regions: 2. Non - blue: 6. So for one spin, probability non - blue is 6/8 = 3/4. For two spins, independent, so (3/4)^2 = 9/16.

Answer:

$\frac{9}{16}$