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.
I believe this is the same question
http://www.jiskha.com/display.cgi?id=1420205675
To solve these questions, we need to use the concept of probability. Let's go through 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 find the probability of getting the first correct answer in the first three questions. Since each question has four choices, the probability of getting the correct answer to any given question by pure guessing is 1/4. Therefore, the probability of getting the first correct answer in the first three questions is (1/4) + (1/4) + (1/4) = 3/4 = 0.75.
ii. To find the probability of getting less than four correct answers in quiz 2, we need to find the probability of getting 0, 1, 2, or 3 correct answers. Since each question has five choices, the probability of getting the correct answer to any given question by pure guessing is 1/5. Therefore, the probability of getting less than four correct answers is P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3).
Let's calculate each term individually:
P(X = 0) = (4/5)^15 ≈ 0.0509
P(X = 1) = 15 * (1/5) * (4/5)^14 ≈ 0.2024
P(X = 2) = (15 choose 2) * (1/5)^2 * (4/5)^13 ≈ 0.3023
P(X = 3) = (15 choose 3) * (1/5)^3 * (4/5)^12 ≈ 0.2685
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) ≈ 0.8239
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. Since Siti is guessing, the probability of getting a correct answer to any given question is 1/4. Therefore, we need to calculate P(X ≥ 6). This probability can be calculated using the binomial formula or by complementing the probability of getting less than six correct answers.
Using the complement rule, P(X ≥ 6) = 1 - P(X < 6) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)]. We've already calculated P(X < 4) in the previous question as approximately 0.8239.
Now let's calculate P(X = 4) and P(X = 5):
P(X = 4) = (8 choose 4) * (1/4)^4 * (3/4)^4 ≈ 0.2373
P(X = 5) = (8 choose 5) * (1/4)^5 * (3/4)^3 ≈ 0.1946
P(X ≥ 6) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)] ≈ 1 - (0.8239 + 0.2373 + 0.1946) ≈ 0.7442
Therefore, there is a probability of approximately 0.7442 that Siti will obtain at least six correct answers and 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 calculate P(Y = 5), where Y represents the number of correct answers in the first seven questions. Since each question has five choices, the probability of getting the correct answer to any given question by pure guessing is 1/5. Therefore, we need to calculate P(Y = 5).
P(Y = 5) = (7 choose 5) * (1/5)^5 * (4/5)^2 ≈ 0.1641
v. Whether this event is considered likely to occur depends on the interpretation of "likely." With a probability of approximately 0.1641, it means there is about a 16.41% chance of obtaining five correct answers before the eighth question in quiz 2. On a relative scale, this may be considered somewhat likely, but it also depends on the context and the specific circumstances.
i. To find the probability that at most three questions must be answered to obtain the first correct answer in quiz 1, we first need to calculate the probability of getting the first correct answer in the first, second, or third question.
Each question has four choices, so the probability of getting the first question correct by pure guessing is 1/4. The probability of not getting it correct is 3/4. Therefore, the probability of needing to answer the first question is (3/4)^0 * (1/4)^1 = 1/4.
Similarly, the probability of needing to answer the second question is (3/4)^1 * (1/4)^1 = 3/16.
The probability of needing to answer the third question is (3/4)^2 * (1/4)^1 = 9/64.
To find the probability that at most three questions must be answered, we add up these probabilities:
P(At most three questions) = P(1st question) + P(2nd question) + P(3rd question)
= 1/4 + 3/16 + 9/64
= 16/64 + 12/64 + 9/64
= 37/64
Interpretation: The probability that at most three questions must be answered to obtain the first correct answer in quiz 1 is 37/64, which means there's a relatively high chance of getting the first correct answer within the first three questions.
ii. For quiz 2, each question has five choices, so the probability of getting a question correct by pure guessing is 1/5. The probability of not getting it correct is 4/5.
To find the probability of less than four correct answers in quiz 2, we need to calculate the probabilities of getting 0, 1, 2, or 3 correct answers.
P(Less than four correct answers) = P(0 correct) + P(1 correct) + P(2 correct) + P(3 correct)
For each value, we calculate the probability of getting that number of correct answers and sum them up.
P(0 correct) = (4/5)^15
P(1 correct) = 15 * (1/5)^1 * (4/5)^14
P(2 correct) = 15 * (2/5)^2 * (3/5)^13
P(3 correct) = 15 * (3/5)^3 * (2/5)^12
Adding these probabilities gives us the final answer.
Interpretation: The probability of less than four correct answers in quiz 2 depends on the values calculated in the steps above. This probability would indicate the chances of a student getting less than four correct answers randomly.
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.
Each question in quiz 1 has four choices, so the probability of getting a question correct by pure guessing is 1/4. The probability of not getting it correct is 3/4.
To calculate the probability of getting at least six correct answers, we need to sum the probabilities of getting 6, 7, or 8 correct answers.
P(At least six correct answers) = P(6 correct) + P(7 correct) + P(8 correct)
Again, we calculate the probabilities for each value and add them up:
P(6 correct) = 8C6 * (1/4)^6 * (3/4)^2
P(7 correct) = 8C7 * (1/4)^7 * (3/4)^1
P(8 correct) = 8C8 * (1/4)^8 * (3/4)^0
Adding these probabilities gives us the final answer.
Interpretation: The probability of obtaining at least six correct answers in quiz 1 determines the likelihood of Siti getting a high score.