Probability: student with 70% of answers

Question

I have this question:

A student takes a test of multiple option where each question has $5$ possible answers. If the student knows the correct answer, he/she selects it; in opposite case, he/she selects one randomly from the possibles $5$. Suppose that the student knows the answer of the $70\%$ of the questions.

-Which is the probability that from one given question the student gives the correct answer?

-If the student gets the correct answer to a question which is the probability that he/she knew that answer?

From the first question I thought that the answer could be $$\frac{7}{10}\cdot P(\text{correct}\mid \text{student doesn't know})=\frac{7}{10}\cdot\frac{\left(\frac{1}{5}\right)}{\left(\frac{3}{10}\right)}=\frac{7}{15}$$

But I don't know if it is correct. Is it? For the second question I don't know how it could be. Could anyone explain me how to tackle them?

Answer 1

$\begin {array}{} &\text {Student Knows}&\text {Student Guesses}&\text {Totals}\\ \text {Correct}& 0.70 &0.30\cdot 0.20 = 0.06&0.76\\ \text {Incorrect}& 0 &0.30\cdot 0.80 = 0.24&0.24\end {array}$

$P(\text {Knows}|\text { Correct}) = \frac {0.70}{0.76}$

Answer 2

No, your answer and approach for the first question is not correct.

Think about it like this:

There are two ways for the student to get the correct answer: either the student knows the answer, or the student does not know, but guessed right. These are mutually exclusive events, and so you should add the probabilities for each of those events to get the probability of getting the correct answer.

So, if we use $C$ for the evnt of the student picking the correct answer, and $K$ for the event of the student knowing the answer, we get:

$$P(C) = P(C \cap K)+ P(C \cap K^C)$$

Also:

$$P(C \cap K) = P(C|K)\cdot P(K)$$

$$P(C \cap K^C) = P(C|K^C)\cdot P(K^C)$$

And we are given that:

$$P(K)=0.7$$

and thus:

$$P(K^C) = 1-0.7=0.3$$

and:

$$P(C|K)=1$$

and

$$P(C|K^C)=0.2$$

So:

$$P(C)= P(C|K)\cdot P(K)+ P(C|K^C)\cdot P(K^C)=1 \cdot 0.7 + 0.2 \cdot 0.3 = 0.76$$

Answer 3

The formula (7/10)(P(correct|student doesn't know)) ignores the possibility that the student knows the answer, and your calculations from there incorrectly evaluates the expression P(correct|student doesn't know).

One strategy is to pick a number of questions such that all the numbers end up being integers. The denominators involved are 5 (five possible answers) and 10 (70% chance knows the answer). So if we multiply those together, we get 50. Draw out a grid with two rows and two columns corresponding to "knows answers"/"doesn't know" and "gets answer right"/"gets answer wrong", and fill out the boxes. Then for each probability asked for, figure out what the population set is, and what the "success" set is, and find the ratio of the two.