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The probability of a man hitting a target is 3/4. He tries 5 times. The probability that the target will be hit at least 3 times, is
- a)291/364
- b)371/464
- c)471/502
- d)459/512
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
The probability of a man hitting a target is 3/4. He tries 5 times. Th...
To find the probability that the target will be hit at least 3 times, we need to consider different scenarios in which the man hits the target 3, 4, or 5 times out of the 5 attempts.
Scenario 1: Hitting the target 3 times
To calculate the probability of hitting the target exactly 3 times, we can use the binomial probability formula. The probability of hitting the target 3 times is given by:
P(3) = (5 choose 3) * (3/4)^3 * (1/4)^(5-3)
Scenario 2: Hitting the target 4 times
Similarly, the probability of hitting the target exactly 4 times can be calculated using the binomial probability formula:
P(4) = (5 choose 4) * (3/4)^4 * (1/4)^(5-4)
Scenario 3: Hitting the target 5 times
The probability of hitting the target all 5 times can be calculated as:
P(5) = (5 choose 5) * (3/4)^5 * (1/4)^(5-5)
Calculating the probabilities
Now, let's calculate the individual probabilities for each scenario:
P(3) = (5 choose 3) * (3/4)^3 * (1/4)^(5-3) = (5!)/(3!2!) * (3/4)^3 * (1/4)^2 = 10 * 27/64 * 1/16 = 270/1024
P(4) = (5 choose 4) * (3/4)^4 * (1/4)^(5-4) = (5!)/(4!1!) * (3/4)^4 * (1/4)^1 = 5 * 81/256 * 1/4 = 405/1024
P(5) = (5 choose 5) * (3/4)^5 * (1/4)^(5-5) = (5!)/(5!0!) * (3/4)^5 * (1/4)^0 = 1 * 243/1024 * 1 = 243/1024
Calculating the probability of hitting the target at least 3 times
To find the probability of hitting the target at least 3 times, we need to sum up the probabilities of scenarios 3, 4, and 5:
P(at least 3) = P(3) + P(4) + P(5) = 270/1024 + 405/1024 + 243/1024 = 918/1024
Simplifying the probability
To simplify the probability, we can divide both the numerator and denominator by their greatest common divisor (GCD), which is 2 in this case:
P(at least 3) = (918/2)/(1024/2) = 459/512
Therefore, the correct answer is option 'D', 459/512.
Scenario 1: Hitting the target 3 times
To calculate the probability of hitting the target exactly 3 times, we can use the binomial probability formula. The probability of hitting the target 3 times is given by:
P(3) = (5 choose 3) * (3/4)^3 * (1/4)^(5-3)
Scenario 2: Hitting the target 4 times
Similarly, the probability of hitting the target exactly 4 times can be calculated using the binomial probability formula:
P(4) = (5 choose 4) * (3/4)^4 * (1/4)^(5-4)
Scenario 3: Hitting the target 5 times
The probability of hitting the target all 5 times can be calculated as:
P(5) = (5 choose 5) * (3/4)^5 * (1/4)^(5-5)
Calculating the probabilities
Now, let's calculate the individual probabilities for each scenario:
P(3) = (5 choose 3) * (3/4)^3 * (1/4)^(5-3) = (5!)/(3!2!) * (3/4)^3 * (1/4)^2 = 10 * 27/64 * 1/16 = 270/1024
P(4) = (5 choose 4) * (3/4)^4 * (1/4)^(5-4) = (5!)/(4!1!) * (3/4)^4 * (1/4)^1 = 5 * 81/256 * 1/4 = 405/1024
P(5) = (5 choose 5) * (3/4)^5 * (1/4)^(5-5) = (5!)/(5!0!) * (3/4)^5 * (1/4)^0 = 1 * 243/1024 * 1 = 243/1024
Calculating the probability of hitting the target at least 3 times
To find the probability of hitting the target at least 3 times, we need to sum up the probabilities of scenarios 3, 4, and 5:
P(at least 3) = P(3) + P(4) + P(5) = 270/1024 + 405/1024 + 243/1024 = 918/1024
Simplifying the probability
To simplify the probability, we can divide both the numerator and denominator by their greatest common divisor (GCD), which is 2 in this case:
P(at least 3) = (918/2)/(1024/2) = 459/512
Therefore, the correct answer is option 'D', 459/512.
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The probability of a man hitting a target is 3/4. He tries 5 times. The probability that the target will be hit at least 3 times, isa)291/364b)371/464c)471/502d)459/512Correct answer is option 'D'. Can you explain this answer?
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