Final answer:
The probability of hitting a target at least twice when firing seven times, with each hit chance being 1/4, is calculated using the complement of the probability of hitting fewer than twice. It involves finding the probability of 0 hits and 1 hit and subtracting their sum from 1.
Step-by-step explanation:
The student is asking about the probability of a man hitting a target at least twice if he fires 7 times, given that the probability of hitting the target once is 1/4. To solve this, we can use the complement rule which states the probability of an event happening at least 'k' times is 1 minus the probability of the event happening fewer than 'k' times. For this problem, we want at least 2 hits, so we look at the probability of 0 or 1 hit.
Let's calculate the probability of 0 hits:
Now, the probability of exactly 1 hit (and 6 misses):
- P(1 hit) = 7 * (1/4) * (3/4)^6
Adding these two probabilities gives us the probability of hitting the target fewer than 2 times.
- P(0 or 1 hit) = (3/4)^7 + 7 * (1/4) * (3/4)^6
The probability of hitting the target at least twice:
- P(at least 2 hits) = 1 - P(0 or 1 hit)
Therefore, the answer is not exactly given in any of the options A, B, C, or D, but the correct method to find the probability of hitting at least twice is through calculation using the above steps.