Sarah took a Statistics midterm which consisted of 13 multiple choice questions. Each question had 6 possible response options. Sarah answered 8 of the 13 questions correctly. Assume that Sarah had the same probability of getting each question correct.
What is the most likely value of p (where p is defined as the probability that Sarah get a given question correct)? You may round the possible values of p to the nearest hundredths place.
With 6 choices for each questions, the probability of getting and question correct = 1/6. Calculate the decimal.
To find the most likely value of p, we need to consider the probability distribution of the number of correct answers Sarah could have obtained.
Given that each question has 6 possible response options and Sarah had the same probability of getting each question correct, the probability of Sarah getting a question correct is p, and the probability of Sarah getting a question incorrect is 1-p.
We can model the number of correct answers Sarah obtained using a binomial distribution. The probability mass function of the binomial distribution is given by:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- X is the random variable representing the number of correct answers,
- n is the total number of questions (13 in this case),
- k is the number of correct answers (8 in this case),
- C(n, k) is the number of combinations of n questions taken k at a time.
To find the most likely value of p, we can calculate the probability of Sarah getting 8 correct answers for different values of p and choose the value that gives us the highest probability.
Let's calculate the probability for different values of p:
When p = 0.1:
P(X=8) = C(13, 8) * 0.1^8 * (1-0.1)^(13-8) = 3.283e-07
When p = 0.2:
P(X=8) = C(13, 8) * 0.2^8 * (1-0.2)^(13-8) = 0.000602
When p = 0.3:
P(X=8) = C(13, 8) * 0.3^8 * (1-0.3)^(13-8) = 0.013369
When p = 0.4:
P(X=8) = C(13, 8) * 0.4^8 * (1-0.4)^(13-8) = 0.076087
When p = 0.5:
P(X=8) = C(13, 8) * 0.5^8 * (1-0.5)^(13-8) = 0.144531
When p = 0.6:
P(X=8) = C(13, 8) * 0.6^8 * (1-0.6)^(13-8) = 0.144531
When p = 0.7:
P(X=8) = C(13, 8) * 0.7^8 * (1-0.7)^(13-8) = 0.076087
When p = 0.8:
P(X=8) = C(13, 8) * 0.8^8 * (1-0.8)^(13-8) = 0.013369
When p = 0.9:
P(X=8) = C(13, 8) * 0.9^8 * (1-0.9)^(13-8) = 0.000602
From the calculations above, we can see that the highest probability occurs when p = 0.5, which gives us a probability of approximately 0.144531.
Therefore, the most likely value of p, rounded to the nearest hundredths place, is 0.14.