An Olympic archer has a 89% chance of hitting a bullseye on a shot. If this archer attempts 6 shots at the target, which of the following is the correct way to calculate the probability of hitting the bullseye no more than four times? Assume all conditions of the binomial distribution are satisfied.

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BlackTom AI · Accepted Answer

First, restating the problem to ensure understanding: we have a binomial scenario with n = 6 trials, probability of success (hitting a bullseye) p = 0.89, and we want the probability of hitting the bullseye no more than four times (i.e., X ≤ 4). Now I will analyze each option one by one.

Option 1: P(x=1) + P(x=2) + P(x=3) + P(x=4)
- This sums the probabilities of getting exactly 1, 2, 3, or 4 successes. Since 'no more than four times' includes 0 as well, this option is missing the term P(x=0). Therefore, it undercounts the desired probability by excluding the case of no bullseyes at all. It is almost correct but incomplete.

Option 2: 1 - P(x=6) - P(x=5)
- Here, we are subtracting the probabilities of getting 6 successes and 5 successes from 1. The complement of 'no more than four successes' is 'five or six successes.' Indeed, the event {X ≤ 4} is the complement of {X ≥ 5} which is {X=5} and {X=6}. Since these two are mutually exclusive and cover all cases with more than four successes, 1 - [P(X=5) + P(X=6)] equals P(X ≤ 4). This reasoning matches the binomial complement rule correctly.

Option 3: 1 - P(x=6) - P(x=5) - P(x=4)
- This expression subtracts the probabilities of 6, 5, and 4 successes from 1. That would give P(X ≤ 3), not P(X ≤ 4). Since it excludes the case of exactly four successes as well, it underestimates the desired probability by omitting P(X=4). Therefore, this is incorrect for the stated goal.

Option 4: P(x=4)
- This option only accounts for the probability of exactly four successes, ignoring all other possibilities (0, 1, 2, 3, and 5 or 6). Since the phrase 'no more than four times' includes all lower counts as well as four, this single-term probability does not capture the full event. It is therefore not correct for the requested calculation.

Summary of the core idea across options: you want P(X ≤ 4). The correct way to obtain this from the binomial distribution is to take the complement of the event X ≥ 5, which is P(X ≤ 4) = 1 - [P(X=5) + P(X=6)]. Among the choices, Option 2 exactly matches this complement approach, while Option 1 misses P(X=0), Option 3 over-subtracts by removing P(X=4) as well, and Option 4 only gives P(X=4) without including the other favorable outcomes.

FA25_QTM_100_3 Final Exam- Requires Respondus LockDown Browser

An Olympic archer has a 89% chance of hitting a bullseye on a shot. If this archer attempts 6 shots at the target, which of the following is the correct way to calculate the probability of hitting the bullseye no more than four times? Assume all conditions of the binomial distribution are satisfied.

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