Richard has just been given a 4-question multiple-choice quiz in his history class. Each question has five answers, of which only one is correct. Since Richard has not attended class recently, he doesn't know any of the answers. Assuming that Richard guesses on all four questions, find the indicated probabilities. (Round your answers to three decimal places.)
(a) What is the probability that he will answer all questions correctly?
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(b) What is the probability that he will answer all questions incorrectly?
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(c) What is the probability that he will answer at least one of the questions correctly? Compute this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table.
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Then use the fact that P(r ≥ 1) = 1 − P(r = 0).
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Compare the two results. Should they be equal? Are they equal? If not, how do you account for the difference?
They should not be equal, but are equal. They should be equal, but may not be due to table error. They should be equal, but may differ slightly due to rounding error. They should be equal, but differ substantially.
(d) What is the probability that Richard will answer at least half the questions correctly?
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0.0039
To calculate the probabilities, we can use the formula for the binomial distribution:
P(x) = nCk * p^k * (1-p)^(n-k)
Where:
- P(x) is the probability of getting x successes
- n is the number of trials (number of questions)
- k is the number of successes (number of correct answers)
- p is the probability of success on a single trial (probability of choosing the correct answer)
In this case,
n = 4 (number of questions)
k = 4 (number of correct answers)
p = 1/5 = 0.2 (probability of choosing the correct answer)
(a) To calculate the probability of answering all questions correctly:
P(4) = 4C4 * (0.2)^4 * (1-0.2)^(4-4)
= 1 * 0.0016 * 0.8^0
= 0.0016
Therefore, the probability of Richard answering all questions correctly is 0.0016.
(b) To calculate the probability of answering all questions incorrectly:
P(0) = 4C0 * (0.2)^0 * (1-0.2)^(4-0)
= 1 * 1 * 0.8^4
= 0.4096
Therefore, the probability of Richard answering all questions incorrectly is 0.4096.
(c) To calculate the probability of answering at least one question correctly using the binomial distribution table:
P(r ≥ 1) = 1 - P(r = 0)
= 1 - P(0)
= 1 - P(4C0 * (0.2)^0 * (1-0.2)^(4-0))
= 1 - P(1 * 1 * 0.8^4)
= 1 - 0.4096
= 0.5904
To calculate the probability of answering at least one question correctly using the fact P(r ≥ 1) = 1 - P(r = 0):
P(r ≥ 1) = 1 - P(0)
= 1 - 0.4096
= 0.5904
The two results are equal, which is what we would expect.
(d) To calculate the probability of Richard answering at least half the questions correctly, we need to find the probabilities of answering 2, 3, and 4 questions correctly and sum them:
P(r ≥ 2) = P(2) + P(3) + P(4)
= 4C2 * (0.2)^2 * (1-0.2)^(4-2) + 4C3 * (0.2)^3 * (1-0.2)^(4-3) + 4C4 * (0.2)^4 * (1-0.2)^(4-4)
= 6 * 0.04 * 0.64 + 4 * 0.008 * 0.8 + 1 * 0.0016 * 0.8^0
= 0.1536 + 0.0256 + 0.0016
= 0.1808
Therefore, the probability of Richard answering at least half the questions correctly is 0.1808.
(a) To find the probability that Richard will answer all questions correctly, we need to find the probability of getting one question correct and multiply it by itself for all four questions.
Since each question has five possible answers, the probability of guessing the correct answer is 1/5. Therefore, the probability of answering one question correctly is 1/5.
To find the probability of answering all four questions correctly, we multiply this probability by itself four times:
P(answer all questions correctly) = (1/5) * (1/5) * (1/5) * (1/5) = 1/625 ≈ 0.0016
Therefore, the probability that Richard will answer all questions correctly is approximately 0.0016.
(b) To find the probability that Richard will answer all questions incorrectly, we need to find the probability of getting one question wrong and multiply it by itself for all four questions.
Since each question has five possible answers and only one is correct, the probability of guessing the wrong answer is 4/5. Therefore, the probability of answering one question incorrectly is 4/5.
To find the probability of answering all four questions incorrectly, we multiply this probability by itself four times:
P(answer all questions incorrectly) = (4/5) * (4/5) * (4/5) * (4/5) ≈ 0.4096
Therefore, the probability that Richard will answer all questions incorrectly is approximately 0.4096.
(c) There are two ways to approach this question. First, using the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table:
Since each question is independent and mutually exclusive, the probability of answering at least one question correctly is equal to 1 minus the probability of answering all questions incorrectly:
P(answer at least one question correctly) = 1 - P(answer all questions incorrectly) = 1 - 0.4096 = 0.5904
Alternatively, we can use the fact that P(r ≥ 1) = 1 − P(r = 0):
P(answer at least one question correctly) = 1 - P(answer no question correctly)
Since the probability of answering each question correctly is 1/5, the probability of not answering any question correctly is (4/5) * (4/5) * (4/5) * (4/5):
P(answer at least one question correctly) = 1 - (4/5) * (4/5) * (4/5) * (4/5) ≈ 0.5904
Both methods yield the same result: the probability that Richard will answer at least one question correctly is approximately 0.5904.
The two results should be equal, and in this case, they are equal.
(d) To find the probability that Richard will answer at least half the questions correctly, we need to consider two cases: he answers 2, 3, or 4 questions correctly.
Case 1: Richard answers 2 questions correctly:
The probability of answering 2 questions correctly is calculated as follows:
P(answer 2 questions correctly) = (1/5) * (1/5) * (4/5) * (4/5) = 16/625 ≈ 0.0256
Case 2: Richard answers 3 questions correctly:
The probability of answering 3 questions correctly is calculated as follows:
P(answer 3 questions correctly) = (1/5) * (1/5) * (1/5) * (4/5) = 4/625 ≈ 0.0064
Case 3: Richard answers 4 questions correctly:
The probability of answering 4 questions correctly is calculated as follows:
P(answer 4 questions correctly) = (1/5) * (1/5) * (1/5) * (1/5) = 1/625 ≈ 0.0016
To find the probability that Richard will answer at least half the questions correctly, we add up the probabilities of these three cases:
P(answer at least half the questions correctly) = P(answer 2 questions correctly) + P(answer 3 questions correctly) + P(answer 4 questions correctly)
= 0.0256 + 0.0064 + 0.0016 = 0.0336
Therefore, the probability that Richard will answer at least half the questions correctly is approximately 0.0336.
a.) 0.0039
b.)0.3164
c.) 0.6836