Beth is taking an eleven-question multiple-choice test for which each question has three answer choices, only one of which is correct. Beth decides on answers by rolling a fair die and marking the first answer choice if the die shows 1 or 2, the second if the die shows 3 or 4, and the third if the die shows 5 or 6. Find the probability of the stated event. exactly four correct answers

Question

Beth is taking an eleven​-question ​multiple-choice test for which each question has three answer​ choices, only one of which is correct. Beth decides on answers by rolling a fair die and marking the first answer choice if the die shows 1 or​ 2, the second if the die shows 3 or​ 4, and the third if the die shows 5 or 6. Find the probability of the stated event.   exactly   four   correct answers

Jenny Bolton · Accepted Answer

Step 1 Introduction: In a fair die, there are 6 possible outcomes, 1, 2, ..., 6, all of which are equally likely, that is, with probability 16. Step 2 Calculation: On rolling the fair die once, the probability of getting 1 or 2 is 13[=16+16=26]. Similarly, the probability of getting 3 or 4 is 1/3, and the probability of getting 5 or 6 is 1/3. Thus, in case of each of the 11 questions, the probability of selecting each option is 1/3. As the outcomes on the roll of a fair die are independent from one roll to another, the choices made in the different questions are also independent of one another. Considering each attempted question as a trial, there are n=11 independent trials. Considering it to be a success if the correct option is chosen, the probability of success in each trial is p=13, as each question has only one correct option. Here, X can be considered as the number of successes, that is, number of questions correctly answered. Thus, X has a binomial distribution with parameters, n=11,p=13. Since p denotes the probability of success, the probability of failure is, q=1−p=23. If X∼Binomial(n,p), then the probability mass function of X is: f(x)={(nx)pxqn−x;x=0,1,...,n;0 Here, n is the number of independent trials, p is the probability of success in each trial, and q is the probability of failure. The probability mass function of X here is: f(x)={(11x)(13)x(23)11−x;x=0,1,2,...,11;0;otherwise The probability of exactly 4 correct answers, that is, exactly 4 successes is calculated below: (X=4)=f(4) =(11x)(13)4(23)11−4 =(330)⋅(181)⋅(1282187) ≈0.238446. Thus, the probability of exactly 4 correct answers is 0.238446.

Debbie Moore · Accepted Answer

Step 1 Binomial Problem with n=10 and p(correct)=13 Find the probability of a) exactly four correct answers P(x=4)=10C4134×236=binopdf(10, 13, 4)=0.2276 b) fewer than three correct answers P(0≤x≤2)=binomcdf(10, 13, 2)=0.2991

nick1337 · Accepted Answer

Step 1 Hence, the probability of getting the correct answer is p=13=0.33333333, as there are 3 choices (the die part simply shows she is guessing). Note that the probability of x successes out of n trials is P(n,x)=nCx px(1−p)n−x where n=number of trials=8 p=the probability of a success=0.333333333 x=the number of successes=6 Thus, the probability is  P(6)=0.017070569

Beth is taking an eleven-question multiple-choice test for

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