1)A quiz consists of 8 multiple choice questions, each with 5 possible answers.
If random guesses are made for all 8 questions, find the probability that a
student will answer exactly 5 correctly.
A. 0.999 B. 0.009 C. 0.990
D. There is not enough information to answer this question.
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d. If random guesses are made for all 8 questions, find the probability that a
student will answer less than 4 correctly.
A. 0.944 B. 0.375 C. 0.046
D. There is not enough information to answer this question.
For each question, prob(correct) = 1/5
prob(wrong) = 4/5
to have exactly 5 of the 8 correct
= C(8,5) (1/5)^5 (4/5)^3
= ....
b) prob(none correct) + prob(1 correct) + prob(2 correct) + prob(3 correct)
= C(8,0) (1/5)^0 (4/5)^8 + C(8,1) (1/5)^1 (4/5)^7 + ...
= appr .16777 + .33554 + ....
= ....
To calculate the probability that a student will answer exactly 5 questions correctly when making random guesses for all 8 questions, we can use the binomial probability formula.
The formula for the probability of exactly x successes in n trials with a probability of success p is:
P(x) = (nCx) * p^x * (1-p)^(n-x)
In this case, there are 8 questions and each question has 5 possible answers, so the probability of answering any question correctly by random guessing is 1/5 = 0.2.
Using the formula, we can calculate the probability of answering exactly 5 questions correctly:
P(5) = (8C5) * (0.2)^5 * (1-0.2)^(8-5)
Calculating this expression gives:
P(5) = (8!/(5!(8-5)!)) * (0.2)^5 * (0.8)^3
P(5) = (8!/(5!3!)) * (0.2)^5 * (0.8)^3
P(5) = (8*7*6)/(3*2*1) * 0.00032 * 0.512
P(5) = 0.0576
Therefore, the probability that a student will answer exactly 5 questions correctly when making random guesses for all 8 questions is 0.0576, which is approximately 0.009. So, the answer is option B: 0.009.
For the second question, to calculate the probability that a student will answer less than 4 questions correctly when making random guesses for all 8 questions, we need to calculate the probabilities of answering 0, 1, 2, or 3 questions correctly and add them together.
P(0) = (8C0) * (0.2)^0 * (0.8)^8
P(1) = (8C1) * (0.2)^1 * (0.8)^7
P(2) = (8C2) * (0.2)^2 * (0.8)^6
P(3) = (8C3) * (0.2)^3 * (0.8)^5
Calculating these expressions gives:
P(0) = (8!/(0!(8-0)!)) * (0.2)^0 * (0.8)^8
P(1) = (8!/(1!(8-1)!)) * (0.2)^1 * (0.8)^7
P(2) = (8!/(2!(8-2)!)) * (0.2)^2 * (0.8)^6
P(3) = (8!/(3!(8-3)!)) * (0.2)^3 * (0.8)^5
P(0) = 0.16777216
P(1) = 0.33554432
P(2) = 0.29360128
P(3) = 0.20575904
Adding these probabilities together gives:
P(<4) = P(0) + P(1) + P(2) + P(3)
P(<4) = 0.16777216 + 0.33554432 + 0.29360128 + 0.20575904
P(<4) = 0.9426768
Therefore, the probability that a student will answer less than 4 questions correctly when making random guesses for all 8 questions is 0.9426768, which is approximately 0.944. So, the answer is option A: 0.944.
To find the probability that a student will answer exactly 5 questions correctly, we can use the binomial distribution formula.
The formula for the probability of getting exactly k successes in n independent Bernoulli trials (where each trial has a probability p of success) is:
P(X = k) = nCk * p^k * (1-p)^(n-k)
In this case, n = 8 (8 questions), k = 5 (exactly 5 correct answers), and p = 1/5 (since there are 5 possible answers for each question and the student is randomly guessing).
Now let's calculate the probability for the first question:
P(X = 5) = 8C5 * (1/5)^5 * (4/5)^3
Using combinations notation, 8C5 = 8! / (5! * (8-5)!), which simplifies to 8! / (5! * 3!).
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
5! = 5 * 4 * 3 * 2 * 1
3! = 3 * 2 * 1
Plugging in these values:
P(X = 5) = (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((5 * 4 * 3 * 2 * 1) * (3 * 2 * 1)) * (1/5)^5 * (4/5)^3
Simplifying the equation:
P(X = 5) = (8 * 7 * 6) / (3 * 2 * 1) * (1/5)^5 * (4/5)^3
P(X = 5) = 0.131
The probability that a student will answer exactly 5 questions correctly is approximately 0.131.
Therefore, the correct answer is not listed among the options given (A, B, C, or D).