# Please explain the steps and results for the following problem. For the following problem, write out the sample space S, give the value of n(S), and tell whether the outcomes are equally likely. Then write the indicated events in parts a–c in set notation. A student must take a 3-question True/False quiz before class over the day’s discussion material. But the student did not read the textbook section and must guess the answer to each question. So each answer is equally likely to be correct or wrong. a) The student gets three answers wrong. b) The students gets exactly two answers correct. c) The student gets only the first answer correct.

You are taking a three question T/F test and guessing on each answer.

The sample space is the set of all possible events, where the events are sets of outcomes. Each question is an outcome and will be either T or F.

The sample space is {TTT, TTF, TFT, FTT, FFT, FTF, TFF, FFF} where n(S)=8. (Each outcome has two possible values, and by the multiplication principle, there are 2*2*2=8 different events.) These are the eight different ways of answering the T/F questions.

(a) The probability that a student gets three answers wrong.

We use the multiplication principle. The probability of three independent events is the product of their probabilities. Here, the chance of missing the first problem is 1/2, as well as the second and third problems. Thus, P=1/2 * 1/2 * 1/2 = 1/8.

(b) The probability that a student gets exactly 2 questions correct.

Let C denote a correct response and W an incorrect response. The possibilities are (CCW), (CWC), and (WCC). In each case, the probability of a correct response, C, is 1/2 and a wrong response, W, is 1/2, and again, we can use the multiplication principle.

P(CCW)=P(CWC)=P(WCC)=1/2 * 1/2 * 1/2 =1/8.

Then we use the addition principal: the probability of three mutually exclusive events is the sum of their probabilities.

1/8+1/8+1/8=3/8.

(c) The student only gets the first answer correct: this happens 1/8 times.

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All correct 1/8 CCC
Exactly 2 correct 3/8 CCW, CWC, WCC
Exactly two wrong 3/8 WWC, WCW, CWW
All incorrect 1/8 WWW

Note that 1/8+3/8+3/8+1/8=1, so we have covered every possibility.