Multiple choice

There are four target balloons. Shooter decides to shoot randomly to hit the target. The probability of hitting a balloon successfully is 1/4. If he is allowed up to four chances to shoot the target, then find the probability that he will hit exactly three balloons.

  1. 3/64

  2. 1/16

  3. 1/8

  4. 3/7

  5. None of the above
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A Correct answer
Explanation

The probability of hitting a balloon is p = 1/4, and missing is q = 3/4. For exactly 3 hits in 4 chances, we use the binomial distribution formula: C(4,3) * (1/4)^3 * (3/4)^1. This equals 4 * (1/64) * (3/4) = 12/256 = 3/64.

AI explanation

Using the binomial distribution formula, the probability of exactly k successes in n trials is P(X=k) = nCk * p^k * q^(n-k). Here, the probability of success p is 1/4, the probability of failure q is 3/4, the number of trials n is 4, and the desired successes k is 3. The calculation is 4C3 times (1/4)^3 times (3/4)^1, which equals 4 times 1/64 times 3/4. Multiplying these values gives 3/64. The probability that he will hit exactly three balloons is 3/64.

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