Final answer:
The man must fire at least 2 times to ensure that the probability of hitting the target at least once is greater than 2/3. To find the number of times the man must fire so that the probability of hitting the target at least once is greater than 2/3, we can use the concept of complementary probability.
Step-by-step explanation:
To find the number of times the man must fire so that the probability of hitting the target at least once is greater than 2/3, we can use the concept of complementary probability. The complementary probability of not hitting the target at least once is equal to 1 minus the probability of hitting the target at least once. Let's represent the number of shots fired as n.
The probability of not hitting the target on a single shot is 1 - 1/4 = 3/4.
The probability of not hitting the target on any of the n shots is (3/4)^n.
The probability of hitting the target at least once is 1 - (3/4)^n.
We need to find the value of n such that the probability of hitting the target at least once is greater than 2/3. So, we have:
1 - (3/4)^n > 2/3
Simplifying the inequality, we get:
(3/4)^n < 1/3
Taking the logarithm of both sides (base 3/4), we have:
n > log((3/4)^n) (base 3/4)
Using logarithmic properties, we can rewrite the inequality as:
n > n * log(3/4) (base 3/4)
Dividing both sides by log(3/4) (base 3/4), we get:
n > 1
Therefore, the man must fire at least 2 times to ensure that the probability of hitting the target at least once is greater than 2/3.