A game of darts involves throwing two darts at the board. Hitting a blue region with a dart scores 1 point; a red region, 2 points, a green region, 3 points, and the orange region, 5 points.
Assume that no skill is involved. What is the probability, stated as a fraction in lowest terms, that two randomly thrown darts will score 8 points or more? Justify your answer.
It depends upon the areas associated with the different colors. You have not said that they are equal.
To calculate the probability of scoring 8 points or more with two randomly thrown darts, we need to consider all the possible combinations that could result in a score of 8 or higher.
First, let's determine the minimum and maximum possible scores with two darts:
- The minimum score is obtained by hitting two blue regions (1 point each), resulting in a score of 2.
- The maximum score is obtained by hitting two orange regions (5 points each), resulting in a score of 10.
Now let's list all the possible combinations of scores that add up to 8 or more:
- 8: A score of 8 is obtained by hitting one red region (2 points) and one blue region (1 point).
- 9: A score of 9 is obtained by hitting one green region (3 points) and one blue region (1 point).
- 10: A score of 10 is obtained by hitting two blue regions (1 point each) or one green region (3 points) and one red region (2 points).
Based on these combinations, there are three ways to obtain a score of 8 or more.
Now, let's determine the total number of possible outcomes. Assuming that no skill is involved, the darts are equally likely to hit any region on the board. The dartboard has a total of four regions: blue, red, green, and orange.
To find the total number of outcomes, we need to consider two darts being thrown at four regions. This can be calculated using the concept of combinations. The total number of possible outcomes is given by the number of ways to select two regions out of four, which can be calculated using the combination formula (nCr):
Total outcomes = 4C2 = 4! / (2!(4-2)!) = 6
Now, we can calculate the probability by dividing the number of favorable outcomes (three ways to score 8 or more) by the total number of outcomes (six):
Probability = Favorable outcomes / Total outcomes = 3/6 = 1/2
Therefore, the probability that two randomly thrown darts will score 8 points or more is 1/2.