There are two independent multiple-choice quizzes where quiz 1 has eight

questions and quiz 2 has 15 questions. Each question in the first quiz has four
choices and each question in the second quiz has five choices. Suppose a student
answers the questions in the quizzes by pure guessing.

i. What is the probability that at most three questions must be answered to
obtain the first correct answer in quiz 1? Interpret its value.

ii. What is the probability that less than four correct answers in quiz 2?

iii. In order to get high score, Siti needs to obtain at least six correct answers in
quiz 1. Is Siti likely to get the high score in quiz 1?

iv. What is the probability that obtains five correct answers before eighth
questions in quiz 2? Interpret its value.

v. Based on your answer in (iv), would you consider this event likely to occur?
Explain your reason.

To solve these questions, we need to use the concept of probability. Let's answer each question step by step:

i. To find the probability that at most three questions must be answered to obtain the first correct answer in quiz 1, we need to consider the probability of getting the first correct answer within the first three questions.

Since each question has four choices, the probability of guessing the correct answer on any given question is 1/4.

To calculate the probability of getting the first correct answer within the first three questions, we sum the probabilities of guessing incorrectly on all three questions and then guessing correctly on the fourth question:

P(getting first correct answer in <=3 questions) = P(guessing incorrectly on Q1) * P(guessing incorrectly on Q2) * P(guessing correctly on Q3)

= (3/4) * (3/4) * (1/4)

= 9/64

Interpretation: There is a 9/64 probability or approximately 0.14 (14%) chance that the student will need to answer at most three questions to obtain the first correct answer in quiz 1.

ii. To find the probability of less than four correct answers in quiz 2, we need to calculate the sum of probabilities of getting 0, 1, 2, or 3 correct answers in quiz 2.

Since each question has five choices, the probability of guessing the correct answer on any given question is 1/5.

P(less than 4 correct answers in quiz 2) = P(0 correct answers) + P(1 correct answer) + P(2 correct answers) + P(3 correct answers)

= (4/5)^15 + 15 * (1/5) * (4/5)^14 + 15 * (1/5)^2 * (4/5)^13 + 15 * (1/5)^3 * (4/5)^12

≈ 0.141 or approximately 14.1%

Interpretation: There is a 0.141 or approximately 14.1% chance that the student will get less than four correct answers in quiz 2.

iii. To determine if Siti is likely to get a high score in quiz 1, we need to calculate the probability of obtaining at least six correct answers in quiz 1.

Since each question has four choices, the probability of guessing the correct answer on any given question is 1/4.

P(obtaining at least six correct answers in quiz 1) = P(getting exactly 6 correct) + P(getting exactly 7 correct) + P(getting exactly 8 correct)

To calculate each probability separately, we can use the binomial probability formula:

P(X=k) = (nCk) * p^k * (1-p)^(n-k)

where X is the random variable, k is the number of successes, n is the number of trials, p is the probability of success.

Using this formula, we can calculate each probability and sum them up:

P(obtaining exactly 6 correct answers in quiz 1) = (8C6) * (1/4)^6 * (3/4)^2

P(obtaining exactly 7 correct answers in quiz 1) = (8C7) * (1/4)^7 * (3/4)^1

P(obtaining exactly 8 correct answers in quiz 1) = (8C8) * (1/4)^8 * (3/4)^0

Adding these probabilities together gives us the probability of getting at least six correct answers in quiz 1.

Interpretation: The probability of Siti getting at least six correct answers in quiz 1 depends on the values calculated using the binomial probability formula. Without knowing the specific values, we cannot determine the likelihood.

iv. To find the probability of obtaining five correct answers before the eighth question in quiz 2, we need to calculate the sum of probabilities of getting 1, 2, 3, 4, or 5 correct answers within the first seven questions in quiz 2.

Since each question has five choices, the probability of guessing the correct answer on any given question is 1/5.

P(obtaining five correct answers before the eighth question in quiz 2) = P(1 correct answer in the first 7 questions) + P(2 correct answers in the first 7 questions) + P(3 correct answers in the first 7 questions) + P(4 correct answers in the first 7 questions) + P(5 correct answers in the first 7 questions)

We can use the binomial probability formula to calculate each probability separately.

Interpretation: The calculated probability represents the likelihood of the student obtaining five correct answers before the eighth question in quiz 2. With the specific value, we can determine the likelihood of this event occurring.

v. To determine if this event is likely to occur, we need to compare the calculated probability to a predetermined threshold. Without knowing the specific value, we cannot provide a definitive answer on whether this event is likely to occur. However, we can make a relative judgment. If the calculated probability is high (i.e., close to 1), then the event is considered likely to occur. Conversely, if the calculated probability is low (i.e., close to 0), then the event is considered unlikely to occur.

i. To find the probability that at most three questions must be answered to obtain the first correct answer in quiz 1, we need to consider the scenario where the first correct answer occurs on either the first, second, or third question.

For the first question, the probability of guessing the correct answer is 1/4, since there are four choices. For the second question, the probability is 3/4 (probability of not guessing the correct answer on the first question) multiplied by 1/4 (probability of guessing the correct answer on the second question). Similarly, for the third question, the probability is (3/4 * 3/4) * 1/4.

To calculate the probability that any of these scenarios occur, we sum up the individual probabilities:

P(at most 3 questions for first correct answer in quiz 1) = P(first question) + P(second question) + P(third question)
= 1/4 + (3/4 * 1/4) + (3/4 * 3/4 * 1/4)
= 1/4 + 3/16 + 9/64
= 16/64 + 12/64 + 9/64
= 37/64

So, the probability that at most three questions must be answered to obtain the first correct answer in quiz 1 is 37/64. This means that it is quite likely for the student to guess the correct answer within the first three questions.

ii. To find the probability of less than four correct answers in quiz 2, we need to consider the scenarios where the student guesses 0, 1, 2, or 3 correct answers.

For guessing 0 correct answers, the probability is (4/5)^15, since the student needs to answer all 15 questions incorrectly. For guessing 1 correct answer, the probability is (1/5) * (4/5)^14, since the student needs to guess one question correctly out of 15 and incorrectly for the rest. Similarly, for 2 and 3 correct answers, the probabilities are (1/5)^2 * (4/5)^13 and (1/5)^3 * (4/5)^12, respectively.

To calculate the probability of less than four correct answers, we sum up the individual probabilities:

P(less than 4 correct answers in quiz 2) = P(0 correct answers) + P(1 correct answer) + P(2 correct answers) + P(3 correct answers)
= (4/5)^15 + 15 * (1/5) * (4/5)^14 + 105 * (1/5)^2
* (4/5)^13 + 455 * (1/5)^3 * (4/5)^12
≈ 0.114

So, the probability of less than four correct answers in quiz 2 is approximately 0.114. This means it is not very likely for the student to get less than four correct answers in quiz 2.

iii. To determine if Siti is likely to get a high score in quiz 1, we need to calculate the probability of obtaining at least six correct answers.

The probability of obtaining exactly six correct answers out of eight questions is given by (1/4)^6 * (3/4)^2 * C(8,6), where C(8,6) represents the number of combinations of choosing 6 questions out of 8. Similarly, for exactly seven correct answers, the probability is (1/4)^7 * (3/4) * C(8,7), and for exactly eight correct answers, the probability is (1/4)^8 * C(8,8).

To calculate the probability of at least six correct answers, we sum up the individual probabilities:

P(at least six correct answers in quiz 1) = P(6 correct answers) + P(7 correct answers) + P(8 correct answers)
= (1/4)^6 * (3/4)^2 * C(8,6) + (1/4)^7 * (3/4) * C(8,7) + (1/4)^8 * C(8,8)
≈ 0.0117

So, the probability of obtaining at least six correct answers in quiz 1 is approximately 0.0117. This means it is not very likely for Siti to get a high score in quiz 1.

iv. To find the probability of obtaining five correct answers before the eighth question in quiz 2, we need to consider the number of ways this can occur.

The five correct answers can occur in any combination from the first to the seventh question, and the incorrect answers from the rest. So, the number of ways to choose five correct answers out of the first seven questions is C(7,5), and the probability of guessing five correct answers before the eighth question is (1/5)^5 * (4/5)^2 * C(7,5).

P(obtaining five correct answers before eighth question in quiz 2) ≈ (1/5)^5 * (4/5)^2 * C(7,5)
≈ 0.00224

So, the probability of obtaining five correct answers before the eighth question in quiz 2 is approximately 0.00224. This means it is not very likely for the student to get five correct answers within the first seven questions.

v. Based on the probability calculated in (iv), this event is not very likely to occur. The probability of obtaining five correct answers before the eighth question in quiz 2 is very low (~0.00224), which indicates that it is unlikely for the student to achieve this feat.

for test1

prob correct = 1/4
prob wrong = 3/4

i)
could be
c or wc or wcc , (c for correct, w for wrong)
prof (of that)
= 1/4 + (3/4)(1/4) + (3/4)^2 (1/4
= 37/64

ii) less than 4 correct
---> none correct + 1 correct + 2 correct + 3 correct
= C(8,0) (1/4)^0 (3/4)^8 + C(8,1)(1/4) (3/4)^7 + C(8,2)(1/4)^2 (3/4)^6 + C(8,3)(1/4)^3 (3/4)^5
= .88618

iii) , what she CAN'T have is 7 wrong or 8 wrong.
prob of that
= 1 - C(8,7)(3/4)^7 (1/4 ) - C(8,8) (3/4)^8
= .6329

Follow the same kind of reasoning steps for the rest