Bayes' Theorem

Q.

In a city$20\%$ people read English newspapers, $40\%$ read Hindi newspapers and $5\%$read both newspapers. The percentage of non-reader either paper is

1. $60\%$

2. $35\%$

3. $25\%$

4. $45\%$

Q. In a test, an examinee either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he makes a guess is $\frac{1}{3}$ and the probability that he copies the answer is $\frac{1}{6}$. The probability that his answer is correct given that he copied it is $\frac{1}{8}.$ Find the probability that he knew the answer to the question, given that he answered it correctly.

1. $\frac{23}{27}$
2. $\frac{17}{21}$
3. $\frac{24}{29}$
4. $\frac{21}{23}$

Q. The probability of selecting a white ball at random from a container that contains only white, yellow and red balls is 1/7. The probability of selecting a yellow ball at random from the same container is 1/5. If the container contains 23 red balls, find the total number of balls in the container.

Q. Given below is the utility schedule of a consumer for commodity-X. The price of the commodity is Rs. 6 per unit. How many units should the consumer purchase to maximise satisfaction? (Assume that utility is expressed in units and 1 util = Rs. 1). Give reasons for your answer.

Consumption (Units)	Total Utility (Utils)	Marginal Utility (Utils)
1	10	10
2	18	8
3	25	7
4	31	6
5	34	3
6	34	0

Q. Let $E_i,i=1,2,...n$ be $n$ independent events such that $P\left(E_i\right)=\frac{1}{i+1}(1\le i\le n)$, the probability at least one of $E_1,E_2,...E_n$ occurs is

1. $\frac{1}{n+1}$
2. $\frac{1}{n!}$
3. $\frac{n}{n+1}$
4. $\frac{n+1}{n!}$

Q. A letter is known to have come either from $TATANAGAR$ or from $CALCUTTA$. On the envelope, just two consecutive letter $TA$ are visible. What is the probability that the letter came from $TATANAGAR$.

1. $\frac{7}{11}$
2. $\frac{2}{3}$
3. $\frac{1}{5}$
4. $\frac{1}{2}$

Q.

$400$ students of class $X$ of a school appeared in a test of $100$ marks in the subject of social
studies and the data about the marks secured is as below :

Marks secured	Number of Students
$0-25$	$50$
$26-50$	$220$
$51-75$	$100$
Above $75$	$30$
Total number of students	$400$

If the result card of a student he picked up at random, what is the probability that the student has secured more than $50$ marks.

1. $0.586$
   
2. $0.325$
3. $0.1$
4. $0.75$

Q. One card is drawn at random form a pack of 52 cards. What is the probability that the card drawn is a king?

1. $\frac{1}{13}$
2. $\frac{1}{52}$
3. $\frac{3}{13}$
4. $\frac{1}{4}$

Q. Two cards are drawn successively with replacement from a well shuffled deck of 52 cards. The probability of drawing two aces is

1. $\frac{1}{13}\times \frac{1}{13}$
2. $\frac{1}{13}\times \frac{1}{17}$
3. $\frac{1}{52}\times \frac{1}{51}$
4. $\frac{1}{13}\times \frac{4}{51}$

Q. The probability that a number selected from the numbers $1,2,3,....,15$ is a multiple of $4$ is $\frac{1}{5}$