Beth is taking an eleven​-question ​multiple-choice test for which each question has three answer​ choices,...

Susan Nall

Susan Nall

Answered

2021-12-17

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

Answer & Explanation

Jenny Bolton

Jenny Bolton

Expert

2021-12-18Added 32 answers

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=1p=23.
If XBinomial(n,p), then the probability mass function of X is:
f(x)={(nx)pxqnx;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)11x;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)114
=(330)(181)(1282187)
0.238446.
Thus, the probability of exactly 4 correct answers is 0.238446.

Debbie Moore

Debbie Moore

Expert

2021-12-19Added 43 answers

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(0x2)=binomcdf(10, 13, 2)=0.2991

nick1337

nick1337

Expert

2021-12-28Added 573 answers

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(1p)nx
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

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