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A student takes a multiple-choice test that has 9 questions. Each question has four choices. The student guesses randomly at each answer. Let X be the number of questions answered correctly. (a) Find P(4)

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Final answer:

To find the probability P(X = 4) of guessing exactly 4 questions correctly on a 9-question multiple-choice test with each question having 4 options, one uses the binomial probability formula with n = 9, k = 4, and p = 0.25, yielding a computed probability based on the calculated number of combinations and the respective powers of success and failure probabilities.

Step-by-step explanation:

The student is dealing with a multiple-choice test where each question has four possible answers, and they are guessing without any prior knowledge. Therefore, the student's chance of correctly guessing any single question is 1 in 4, or 0.25. This scenario can be modeled using the binomial probability distribution, where the random variable X represents the number of questions answered correctly.

To find P(X = 4), which is the probability that the student guesses exactly 4 questions correctly out of 9, we use the binomial probability formula:

P(X = k) = C(n, k) × p^k × (1-p)^(n-k)

Where:

  • n is the number of trials (in this case, 9 questions).
  • k is the number of successes (correct answers, which we want to be 4).
  • p is the probability of success on a single trial (0.25 for each question).
  • C(n, k) is the number of combinations of n items taken k at a time.

We calculate:

P(X = 4) = C(9, 4) × (0.25)^4 × (0.75)^5

The number of combinations C(9, 4) can be calculated as 9!/(4! × (9-4)!), which equals 126. Plugging the values into the formula yields:

P(X = 4) = 126 × (0.25)^4 × (0.75)^5

Completing the calculation provides the probability of the student answering exactly 4 questions correctly by guessing.

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