In an examination a candidate is given the four answers to four questions but is not told which answer applies to which question. He is asked to write down each of the four answers next to its appropriate question.

b). show that there are 6 ways of getting just two answers in the correct places.

c). If a candidate guesses at random where the four answers are to go and X is the number of correct guesses he makes, draw up the probability distribution for X in tabular form.

d). Calculate the expectation of X.

3 answers

Bnk

24 To solve this problem, we need to use concepts from probability and combinatorics. Let's break down each part step by step:

b). To show that there are 6 ways of getting just two answers in the correct places, we can consider the number of ways to select two questions out of the four to have the correct answer. This can be done using the combination formula.

The number of ways to choose 2 questions out of 4 is given by C(4, 2) = 4! / (2!(4-2)!) = 6.

Thus, there are 6 ways of getting just two answers in the correct places.

c). To draw up the probability distribution for X (the number of correct guesses the candidate makes), we need to consider all possible outcomes and their corresponding probabilities.

Let's assume that the candidate randomly assigns the answers to the questions. For each question, there are 4 choices, so the total number of possible outcomes is 4^4 = 256.

Now, we need to determine the number of ways the candidate can make different numbers of correct guesses. If the candidate makes X correct guesses, then he must also make (4-X) incorrect guesses.

The probability of getting X correct guesses is given by:

P(X correct guesses) = C(4, X) * (1/4)^X * (3/4)^(4-X)

Here, C(4, X) is the number of ways to choose X questions out of 4, (1/4)^X is the probability of getting X correct guesses, and (3/4)^(4-X) is the probability of getting (4-X) incorrect guesses.

Using this formula, we can calculate the probability distribution for X by substituting different values of X, ranging from 0 to 4, into the formula.

d). To calculate the expectation of X, we need to multiply each value of X by its corresponding probability and sum them up. The formula for expectation is:

Expectation of X = Σ(X * P(X correct guesses))

Here, the summation is done over all possible values of X, from 0 to 4. Substitute the calculated probabilities for each value of X into the formula and sum them up to find the expectation of X.