Answer:
a. Since blue occurs a fraction z of the time and the red and green portions have equal area, the probability of landing on green is (1 - z)/2.
b. The area diagram for spinning the spinner twice would have three sections: blue, green, and red. Each section would be divided into three smaller sections: BB, BG, BR, GB, GG, GR, RB, RG, and RR. The diagram would show all 9 possible outcomes of spinning the spinner twice.
c. The region on the area diagram corresponding to getting the same color on the spinner twice would be the diagonal line from the upper left to the lower right, going through the GG, BB, and RR sections.
d. The probability that both spins give the same color can be calculated as follows:
- The probability of getting blue on the first spin is z.
- The probability of getting green on the first spin is (1 - z)/2.
- The probability of getting red on the first spin is (1 - z)/2.
- If the first spin is blue, the probability of getting blue again on the second spin is z. The same applies to green and red.
- Therefore, the probability of getting the same color on both spins is z^2 + ((1 - z)/2)^2 + ((1 - z)/2)^2 = z^2 + (1 - z)^2/4.
e. If you know that you got the same color twice, the probability that the color was blue can be calculated using Bayes' theorem:
- Let A be the event that you got blue twice, and let B be the event that you got the same color twice.
- Then P(A|B) = P(B|A) * P(A) / P(B), where P(A) = z^2, P(B|A) = 1, and P(B) = z^2 + ((1 - z)/2)^2 + ((1 - z)/2)^2 = z^2 + (1 - z)^2/4.
- Therefore, P(A|B) = z^2 / (z^2 + (1 - z)^2/4).
Explanation:
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