Multiple-choice questions each have five possible answers left parenthesis a comma b comma c comma d comma e right parenthesis, one of which is correct. Assume that you guess the answers to three such questions.
a. Use the multiplication rule to find P(WWC), where C denotes a correct answer and W denotes a wrong answer.
P(WWC)equals
nothing (Type an exact answer.)
b. Beginning with WWC, make a complete list of the different possible arrangements of one correct answer and two wrong answers, then find the probability for each entry in the list.
P(WWC)minussee above
P(WCW)equals
P(CWW)equals (Type exact answers.)
c. Based on the preceding results, what is the probability of getting exactly one correct answer when three guesses are made?
nothing (Type an exact answer.)
a. To find P(WWC), we use the multiplication rule. There is a 4/5 chance of getting the first question wrong, a 4/5 chance of getting the second question wrong, and a 1/5 chance of getting the third question correct. Multiply these probabilities together:
P(WWC) = (4/5) * (4/5) * (1/5) = 16/125
b. There are three different possible arrangements of one correct answer and two wrong answers: WWC, WCW, and CWW. We have already found P(WWC) in part (a). Now we need to find P(WCW) and P(CWW):
P(WCW) = (4/5) * (1/5) * (4/5) = 16/125
P(CWW) = (1/5) * (4/5) * (4/5) = 16/125
c. To find the probability of getting exactly one correct answer when three guesses are made, sum up the probabilities found in part (b):
P(Exactly one correct answer) = P(WWC) + P(WCW) + P(CWW) = 16/125 + 16/125 + 16/125 = 48/125
a. To find the probability of getting WWC (one correct answer followed by two wrong answers), we use the multiplication rule.
Since each question has 5 possible answers, the probability of guessing a correct answer is 1/5 and the probability of guessing a wrong answer is 4/5.
Therefore,
P(C) = 1/5 (probability of guessing a correct answer)
P(W) = 4/5 (probability of guessing a wrong answer)
Using the multiplication rule, we multiply the probabilities together:
P(WWC) = P(W) * P(W) * P(C) = (4/5) * (4/5) * (1/5) = 16/125
b. To find the different possible arrangements of one correct answer and two wrong answers (WWC, WCW, CWW) and their probabilities, we use the multiplication rule again.
For WWC:
P(WWC) = (4/5) * (4/5) * (1/5) = 16/125
For WCW:
P(WCW) = (4/5) * (1/5) * (4/5) = 16/125
For CWW:
P(CWW) = (1/5) * (4/5) * (4/5) = 16/125
c. To find the probability of exactly one correct answer when three guesses are made, we sum up the probabilities of getting one correct answer in each possible arrangement (WWC, WCW, CWW).
P(exactly one correct answer) = P(WWC) + P(WCW) + P(CWW)
= (16/125) + (16/125) + (16/125)
= 48/125
a. To find P(WWC), we need to use the multiplication rule. The multiplication rule states that the probability of two independent events occurring together is the product of their individual probabilities. In this case, we have three independent events: guessing the correct answer (C) or the wrong answer (W) in three different questions.
Since there is only one correct answer and four wrong answers (W, X, Y, Z) in each question, the probability of guessing a correct answer is 1/5, and the probability of guessing a wrong answer is 4/5.
So, the probability of getting a wrong answer (W) followed by another wrong answer (W) and then a correct answer (C) is:
P(WWC) = P(W) * P(W) * P(C)
= (4/5) * (4/5) * (1/5)
= 16/125
Therefore, P(WWC) = 16/125.
b. To find the probability of different arrangements of one correct answer (C) and two wrong answers (W) out of three guesses, we need to consider all possible arrangements. The possible arrangements are:
WWC, WCW, CWW
For each arrangement, we use the same probabilities as before: P(C) = 1/5, and P(W) = 4/5.
So, the probabilities for each arrangement are:
P(WWC) = (4/5) * (4/5) * (1/5) = 16/125
P(WCW) = (4/5) * (1/5) * (4/5) = 16/125
P(CWW) = (1/5) * (4/5) * (4/5) = 16/125
Therefore, P(WCW) = P(CWW) = 16/125.
c. To find the probability of getting exactly one correct answer when three guesses are made, we need to add the individual probabilities of each arrangement where only one guess is correct. In this case, it would be the probabilities of getting one correct answer out of three, which are:
P(WCW) = 16/125
P(CWW) = 16/125
Adding these probabilities:
P(exactly one correct answer) = P(WCW) + P(CWW) = 16/125 + 16/125
= 32/125
Therefore, the probability of getting exactly one correct answer when three guesses are made is 32/125.