Question 1 :
A purse contains $2$ six-sided dice. One is a normal fair die,while the other has two $1's, $ two $ 3' s$ and two $5'$, A die is picked up and rolled. Because of some secret magnetic attration of the unfair die, there is 75% chance of picking the unfair die and a 25% chance of picking a fair die. The die is rolled and shows up the face $3$ The probability that a fair die was picked up is

Question 2 :
There are two urns. There are $m$ white & $n$ black balls in the first urn and $p$ white & $q$ black balls in the second urn. One ball is taken from the first urn & placed into the second. Now, the probability of drawing a white ball from the second urn is

Question 3 :
Assertion: $ P\left ( H/E \right )> P\left ( E/H_{i} \right )P\left ( H_{i} \right ),\:i=1,\:2,\:3,...,n\: $.Let $ H_{1},H_{2},H_{3},.....H_{n} $ be n mutually exclusive & exhaustive events with probability $ P\left ( H_{i} \right )> 0,i=1,2,3,...n $. Let E be any other event with $ 0 < P \left ( E \right )< 1 $ Reason: $ \sum_{i=1}^{n}P\left ( H_{i} \right )= 1 $

Question 4 :
Two events $A$ and $B$ are such that $P(A)=\cfrac{1}{4},P(B)=\cfrac{1}{2}$ and $P(B| A)=\cfrac{1}{2}$<br/>Consider the following statements<br/>$(I)$ $P(\overline { A } |\overline { B } )=\cfrac { 3 }{ 4 } $<br/>$(II)$ $A$ and $B$ are mutually exclusive<br/>$(III)$ $P(\overline { A } |\overline { B } )+P(A|\overline { B } )=1$<br/>Then

Question 5 :
A certain party consists of four different group of people - 30 students, 35 politicians, 20 actors and 27 leaders. On a particular function day, the total cost spent on party members was Rs. 9000. It was found that 6 students spent as much as 7 politicians, 15 politicians spent as much as 12 actors and 10 actors spent as much as 9 leaders. How much did students spent ?

Question 6 :
Assertion: If $ P\left ( A/B \right )> P\left ( A \right ) $ then. $ P\left ( B/A \right )> P\left ( B \right ) $. Reason: If events A & B are dependent, then $ P\left ( A/B \right )= \displaystyle \frac{P\left ( A\cap B \right )}{P\left ( B \right )} $.

Question 7 :
A fair coin is tossed five times. If the out comes are $2$ heads and $3$ tails (in some order), then what is the probability that the fourth toss is a head?

Question 8 :
A candidate takes three tests in succession and the probability of passing the first test is P. The probability of passing each succeeding test is P or $\frac{P}{2}$ according as he passes or fails in the preceding one. The candidate is selected if he passes at least two tests. The probability that the cndidate is selected is

Question 9 :
If the integers $m$ and $n$ are chosen at random from $1$ to $100$ then the probability that ${7}^{m}+{7}^{n}$ is divisible by $5$ is&#160; ?<br/>

Question 10 :
In a set of $10$ coins, $2$ coins are with heads on both the sides. A coin is selected at random from this set and tossed five times. If all the five times, the result was heads, find the probability that the selected coin had heads on both the sides.&#160;&#160;<br/>&#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160;&#160;<br/>