When Ayla improves her game of darts the chances that a she hits a bullseye is 0.65. Assume that each throw is independent.
A) what are the chances that three darts fired in succession will all hit Bullseye.
B) what is the probability that none will hit
C) what is the probability that at least one will hit?
D) what is the probability that two will hit
A) Since each throw is independent, the probability that three darts fired in succession will all hit Bullseye is the product of the individual probabilities. Therefore, the probability is 0.65 * 0.65 * 0.65 = 0.274625.
B) The probability that none will hit is the complement of the probability that at least one will hit. Therefore, the probability is 1 - P(at least one will hit).
C) To find the probability that at least one will hit, we can use the complement rule again. The probability that at least one will hit is equal to 1 - P(none will hit). Since there are three darts fired, the probability that none will hit is (1-0.65)^3 = 0.0359375. Therefore, the probability that at least one will hit is 1 - 0.0359375 = 0.9640625.
D) The probability that two will hit is the probability that the first dart hits Bullseye (0.65) multiplied by the probability that the second dart hits Bullseye (0.65) multiplied by the probability that the third dart does not hit Bullseye (1 - 0.65). Therefore, the probability is 0.65 * 0.65 * (1 - 0.65) = 0.225125.
To calculate the probabilities, we will use the formula:
P(success) = probability of success
A) The chances that three darts fired in succession will all hit the bullseye can be calculated by multiplying the probabilities of each individual throw:
P(all hit) = P(success) * P(success) * P(success)
Substituting the given probability:
P(all hit) = 0.65 * 0.65 * 0.65 = 0.274625
So, the chances that three darts fired in succession will all hit bullseye is approximately 0.2746.
B) The probability that none of the darts hit the bullseye is equal to the complement of the probability that all hit:
P(none hit) = 1 - P(all hit)
Substituting the previously calculated result:
P(none hit) = 1 - 0.274625 = 0.725375
So, the probability that none of the darts hit the bullseye is approximately 0.7254.
C) The probability that at least one dart hits the bullseye is equal to the complement of the probability that none hit:
P(at least one hit) = 1 - P(none hit)
Substituting the previously calculated result:
P(at least one hit) = 1 - 0.725375 = 0.274625
So, the probability that at least one dart hits the bullseye is approximately 0.2746.
D) The chances of exactly two darts hitting the bullseye can be calculated using a combination of the probabilities of success and failure:
P(two hit) = (P(success) * P(success) * P(failure)) + (P(success) * P(failure) * P(success)) + (P(failure) * P(success) * P(success))
Substituting the given probability:
P(two hit) = (0.65 * 0.65 * (1 - 0.65)) + (0.65 * (1 - 0.65) * 0.65) + ((1 - 0.65) * 0.65 * 0.65) = 0.4700625
So, the probability that two darts hit the bullseye is approximately 0.4701.