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.