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If the dartboard is hit, the probability that region A, B and C are hit are proportional to their respective areas. r1, r2 and r3 represent the radius of the three concentric circles and r1: r2: r3 = 1:2:3. A score of 3 ,2 and 1 are obtained if region A, B and C are hit respectively. John threw two darts to the dartboard randomly and both darts hit the dartboard.What is the probability of getting a score of at least 4?

Question

If the dartboard is hit, the probability that region A, B and C are hit are proportional to their respective areas. r1, r2 and r3 represent the radius of the three concentric circles and r1: r2: r3 = 1:2:3. A score of 3 ,2 and 1 are obtained if region A, B and C are hit respectively. John threw two darts to the dartboard randomly and both darts hit the dartboard.What is the probability of getting a score of at least 4?

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Solution

To solve this problem, we first need to understand the areas of the regions A, B, and C. Since the radii are in the ratio 1:2:3, the areas (which are proportional to the square of the radius) will be in the ratio 1^2:2^2:3^2 = 1:4:9.

Next, we need to calculate the probabilities of hitting each region. The total area is 1+4+9 = 14. So, the probabilities of hitting regions A, B, and C are 1/14, 4/14, and 9/14 respectively.

Now, let's consider the ways to score at least 4:

  1. Two darts hit region A (score = 3*2 = 6). The probability of this happening is (1/14)^2 = 1/196.
  2. One dart hits region A (score = 3) and the other hits region B (score = 2). The probability of this happening is 2*(1/14)*(4/14) = 8/196.
  3. One dart hits region A (score = 3) and the other hits region C (score = 1). The probability of this happening is 2*(1/14)*(9/14) = 18/196.
  4. Two darts hit region B (score = 2*2 = 4). The probability of this happening is (4/14)^2 = 16/196.

Adding these probabilities together, we get the total probability of scoring at least 4 as 1/196 + 8/196 + 18/196 + 16/196 = 43/196.

This problem has been solved

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