In An Examination Has 100 Questions. A student Has 60% Chance Of Getting Each Question Correct. Astudent Fails The Examination For Amark Less Than 55. A Student Gets Distinction For A mark Of 68 Or More. Calculate That A Student:
(i) Fails The Examination,
(ii) Gets A Distinction.
To calculate the probabilities of a student failing the examination or getting a distinction, we need to consider the probabilities of getting a certain number of questions correct.
(i) Calculating the probability of failing the examination:
To find the probability of a student failing the examination, we must calculate the probability of them answering fewer than 55 questions correctly.
The student has a 60% chance of getting each question correct, so they have a 40% chance of getting it wrong. The probability of answering a question incorrectly is 1 - 0.60 = 0.40.
Let's calculate the probability of getting fewer than 55 questions correct:
P(fail) = P(x < 55)
P(fail) = P(x = 0) + P(x = 1) + ... + P(x = 54)
Using the binomial probability formula, we can calculate the probability for each number of correct answers and sum them up.
P(x = k) = C(100, k) * (0.40)^(100 - k) * (0.60)^k
Where C(100, k) is the combination of 100 items taken k at a time, which can be calculated as C(100, k) = 100! / (k! * (100 - k)!)
By summing up all these probabilities, we'll get the probability of failing the examination.
(ii) Calculating the probability of getting a distinction:
To find the probability of a student getting a distinction, we must calculate the probability of them answering 68 or more questions correctly.
Let's calculate the probability of getting 68 or more questions correct:
P(distinction) = P(x ≥ 68)
P(distinction) = P(x = 68) + P(x = 69) + ... + P(x = 100)
Again, using the binomial probability formula, we can calculate the probability for each number of correct answers and sum them up.
P(x = k) = C(100, k) * (0.40)^(100 - k) * (0.60)^k
By summing up all these probabilities, we'll get the probability of getting a distinction.
Please note that these calculations may involve large numbers and require computational tools like a calculator or a programming language.