If $P(A) = 0.8$, $P(B) = 0.5$ and $P(B|A) = 0.4$, find (i) $P(A \cap B)$
2.
The probability of obtaining an even prime number on each die, when a pair of dice is rolled is
a.
0
b.
$\frac{1}{3}$
c.
$\frac{1}{12}$
d.
$\frac{1}{36}$
3.
A fair die is rolled. Consider events $E = \{1,3,5\}$, $F = \{2,3\}$ and $G = \{2,3,4,5\}$. Find (i) $P(E|F)$ and $P(F|E)$
4.
A die is tossed thrice. Find the probability of getting an odd number at least once.
5.
A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.
6.
A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?
7.
$A$ and $B$ are two events such that $P(A) \neq 0$. Find $P(B|A)$, if (i) $A$ is a subset of $B$
8.
Two events A and B will be independent, if
a.
A and B are mutually exclusive
b.
$P(A'B') = [1 – P(A)] [1 – P(B)]$
c.
$P(A) = P(B)$
d.
$P(A) + P(B) = 1$
9.
If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability $\frac{1}{2}$).
10.
Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that (ii) first ball is black and second is red.
11.
Determine $P(E|F)$. Two coins are tossed once, where (ii) E : no tail appears, F : no head appears
12.
Given that the events A and B are such that $P(A) = \frac{1}{2}$, $P(A \cup B) = \frac{3}{5}$ and $P(B) = p$. Find $p$ if they are (i) mutually exclusive
13.
Given that the two numbers appearing on throwing two dice are different. Find the probability of the event ‘the sum of numbers on the dice is 4’.
14.
Determine $P(E|F)$. Two coins are tossed once, where (i) E : tail appears on one coin, F : one coin shows head
15.
Given two independent events A and B such that $P(A) = 0.3$, $P(B) = 0.6$. Find (i) P(A and B)
16.
Events A and B are such that $P(A) = \frac{1}{2}$, $P(B) = \frac{7}{12}$ and P(not A or not B) = $\frac{1}{4}$. State whether A and B are independent ?
17.
Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?
18.
An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?
19.
Evaluate $P(A \cup B)$, if $2P(A) = P(B) = \frac{5}{13}$ and $P(A|B) = \frac{2}{5}$
20.
If A and B are any two events such that $P(A) + P(B) – P(A \text{ and } B) = P(A)$, then
a.
$P(B|A) = 1$
b.
$P(A|B) = 1$
c.
$P(B|A) = 0$
d.
$P(A|B) = 0$
