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gayaneshka [121]
3 years ago
7

In answering on a multiple choice test, a student either know the answer or guesses. Let p be the probability that the students

knows the answer and 1-p be the probability that the student guesses. Assume that a student who guesses at the answer will be correct with probability 1/m, where m is the number of multiple choice alternatives. What is the conditional probability that a student knew the answer to a question when he or she answered it correctly?
Mathematics
1 answer:
LenaWriter [7]3 years ago
5 0

Answer:

P(A_{1}|B ) =\frac{mp}{1+p(m-1)}

Step-by-step explanation:

For mutually exclusive events as A1, A2, A3, etc, Bayes' theorem states:

P(A|B)= \frac{P(B|A)P(A)}{P(B)}

P(A|B) is a conditional probability: the likelihood of event A occurring given that B is true.

P(B|A) is a conditional probability: the likelihood of event B occurring given that A is true.

P(A) is the probability that A occurs

P(B) is the probability that B occurs

For this problem:

A1 is the probability that the student knows the answer

A2 is the probability that the student guesses the answer

B is the probability that the student answer correctly

P(A_{1})=p \\P(A_{2})=1-p \\P(B|A_{1})=1 \\P(B|A_{2})=\frac{1}{m} \\P(B)= P(A_{1})P(B|A_{1}) + P(A_{2})P(B|A_{2})= p+\frac{1-p}{m} \\

P(B|A₁) means the probability that the answer is correct when he knew the answer

P(B|A₂) means the probability that the answer is correct when he guessed the answer

P(A₁|B) means the probability that he knew the answer when the answer was correct

Replacing everything in the Bayes' theorem you get:

P(A_{1}|B)= \frac{P(B|A_{1})P(A_{1})}{P(B)}=\frac{(1)(p)}{p+\frac{1-p}{m}} =\frac{mp}{mp+1-p} =\frac{mp}{1+p(m-1)}

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The circumference of Circle K is pi. The circumference of Circle L is 4xpi.
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Answer:

Ratio of circumferences: \displaystyle\frac{1}{4}

Ratio of radii: \displaystyle\frac{1}{4}

Ratio of areas: \displaystyle\frac{1}{16}

Step-by-step explanation:

Hi there!

We are given:

- The circumference of Circle K is \pi

- The circumference of Circle L is 4\pi

Therefore, the ratio of their circumferences would be:

\displaystyle\frac{\pi}{4\pi} ⇒ \displaystyle\frac{1}{4} when simplified

The formula for circumference is C=2\pi r, where <em>r</em> is the radius. To find the ratio of the circles' radii, we must identify their radii through their given circumferences.

If the circumference of Circle K is \pi, or 1\pi, then its radius is \displaystyle\frac{1}{2}.

If the circumference of Circle L is 4\pi, then its radius is \displaystyle\frac{4}{2}, which is 2.

Therefore the ratio their radii would be:

\displaystyle\frac{\frac{1}{2}}{{2}} ⇒ \displaystyle\frac{1}{2}*\frac{1}{2} ⇒ \displaystyle\frac{1}{4} when simplified

The formula for area is:

A=\pi r^2

First, let's find the area of Circle K:

A=\pi (\displaystyle\frac{1}{2})^2\\\\A=\displaystyle\frac{1}{4}\pi

Now, let's find the area of Circle L:

A=\pi (2)^2\\A = 4\pi

Therefore, the ratio of their areas would be:

\displaystyle\frac{\frac{1}{4}\pi}{4\pi} ⇒ \displaystyle\frac{\frac{1}{4}}{4} ⇒ \displaystyle\frac{1}{4} * \frac{1}{4} ⇒ \displaystyle\frac{1}{16} when simplified

I hope this helps!

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