PROBABILITY

ARPIT CHOUDHRY
SIR
What is the probability of throwing a number greater than 3 with an ordinary die?
Two coins are tossed. What is the probability of getting (i) one head (ii) atleast
one head.
In a single throw of two dice, find the probability of getting
(i) a total of 8
(ii) a total of 11
(iii) a total of 3 or 5
(iv) getting the sum an odd number
(v) not getting the same number on the two dice.
Find the probability of getting the sum a prime number when two dice are
thrown together.
Three coins are tossed once. Find the probability of getting
(i) 3 heads
(ii) exactly 2 heads
(iii) atleast 2 heads
(iv) atmost 2 heads
(v) no heads
(vi) exactly 2 tails
(vii) exactly one tail
(viii) a head on the first coin.
A coin and a die are thrown. what is the probability of getting
(i) a head
(ii) an odd number
(iii) a head and an even number.
One card is drawn from a well shuffled deck of 52 cards. If each outcome is
equally likely, calculate the probability that the card will be
(i) a heart
(ii) not a heart
(iii) a black card (i.e., a club or a spade)
(iv) not a black card
(v) not an ace
(vi) an ace of spade.
In a class of 25 students with roll numbers 1 to 25 , a student is picked up at random
to answer a question. Find the probability that the roll number of the selected
student is either a multiple of 5 or of 7.
A die is thrown, find the probability of following events :
(i) A prime number will appear
(ii) A number greater than or equal to 3 will appear
(iii) A number less than or equal to one will appear
(iv) A number more than 6 will appear
(v) A number less than 6 will appear.
20 cards are numbered from 1 to 20. One card is then drawn at
random. What is the probability that the number on the card will be
(i) a multiple of 4
(ii) odd
(iii) greater than 12
(iv) divisible by 5
(v) not a multiple of 6 .
From a pack of 52 cards, two cards are drawn at random. Find the
(i) probability of drawing two aces.
(ii) probability that one is king and the other is queen.
(iii) probability of getting two cards of the same suit.
In a lottery, a person chooses six different natural numbers at random from 1 to
20 , and if these six numbers match with the six numbers already fixed by the
lottery committee, he wins the prize. What is the probability of winning the
prize in the game?
Out of 21 tickets marked with numbers 1 to 21, three are drawn at random.
Find the probability that the numbers on them are in A.P.
A bag contains 30 tickets, numbered from 1 to 30. Five tickets are drawn at random
and arranged in ascending order. Find the probability that the third number is 20.
Four digit numbers are formed by using the digits 1, 2, 3, 4 and 5 without
repeating any digit. Find the probability that a number; chosen at random, is an
add number.
Find the probability that in a random arrangement of letters of the word
'UNIVERSITY', two 'I's do not come together.
A card is drawn at random from a well shuffled pack of 52 cards. Find the
probability of getting :
(i) a two of heart or diamond
(ii) a jack or a queen or a king.
In a given race, the odds in favour of three horses, are and
respectively. Assuming that a dead heat is impossible, find the probability that
one of them wins.
A and are two non-mutually exclusive events. If and
, find the value of and .
Given and ; find
(i)
(ii)
(iii)
(iv)
where denotes the events not and not respectively.
A number is chosen at random from numbers from 1 to 50. What is the
probability that the number is a multiple of 2 or 3.
The probability that a student will pass the final examination in both English and
Hindi is 0.5 and the probability of passing neither is 0.1. If the probability of
passing the English examination is 0.75, what is the probability of passing the
Hindi examination?
[NCERT]
If and are two mutually exclusive events in a sample space such that
and , then find
(i)
(ii)
(iii)
(iv) .
and are two mutually exclusive events of an experiment. If ,
and , find the value of .
If and are mutually exclusive exhaustive events such that ; find
.
A, B, C are three mutually exclusive and exhaustive events associated with a
random experiment. Find , given that and .
A card is drawn from a pack of playing cards. What is the probability that it will
be either a king of diamond or a queen of heart?
A bag contains 6 white, 5 black and 4 yellow balls. Find the chance of getting either
a white or a black ball in a single draw.
What is the chance of throwing a total of 5 or 6 in a single throw of two dice ?
In a single throw of two dice, what is the probability that the sum of the numbers
on the two faces is either 9 or 11 (ii) neither 9 nor 11 .
In a race, the odds in favour of four horses A, B, C, D are and
respectively. Assuming that a dead heat (not more than one wins at a time) is
impossible, find the chance that one of them will win the race. If there are five
horses in all, what is the probability of success for the fifth horse?
There are three events A, B, C one of which must happen and only one can happen
at a time. If the odds are 8 to 3 against A, 5 to 2 against ; find the odds against C.
There are 3 red and 2 black balls in a bag. 3 balls are taken out at random from
the bag. Find the probability of getting 2 red and 1 black or 1 red and 2 black balls.
In a group of students, there are 3 boys and 3 girls. Four students are to be selected at
random from the group. Find the probability that either 3 boys and 1 girl, or 3 girls
and 1 boy are selected.
Two cards are drawn from a pack of cards. What is the chance that atleast one of
the cards drawn is an ace.
Two cards are drawn at random from 8 cards numbered from 1 to 8 . What is
the probability that the sum of the numbers is even odd, if the two
cards are drawn together.
CONDITIONAL PROBABILITY
If E and F are two events such that.
and ; find and .
Given and . Find and .
If , then find
A coin is tossed twice and the four possible outcomes are assumed to be
equally likely. If is the event: "both head and tail have occurred" and is
the event "atmost one tail is observed", find .
A pair of fair dice is thrown. Find the probability that the sum is 10 or greater if
5 appears on first die.
A die is thrown twice and the sum of numbers appearing is observed to be .
What is the conditional probability that number 2 has appeared at least once.
A die is thrown three times. Events and are defined as below:
A: 4 on the third throw
on the first and 5 on the second throw
Find the probability of A given that has already occurred.
In a certain college, of the students failed in Mathematics, of the
students failed in Chemistry and failed in both Mathematics and
Chemistry. A student is selected at random. What is the probability that he
has failed in Mathematics given that he has failed in Chemistry? What
values should a student possess so that he does not fail ?:
In a school there are 1000 students, out of which 430 are girls. It is known
that out of of the girls study in Class XII. What is the probability
that a student chosen randomly studies in Class XII, given that the chosen
student is a girl?
In a girls hostel 60 % of the students read Hindi newspaper: 40% read English
newspaper and read both Hindi and English newspapers. A student is
selected at random.
(i) Find the probability that she reads neither Hindi nor English newspapers.
(ii) If she reads Hindi newspaper, find the probability that she reads English
newspaper.
(iii) If she reads English newspaper, find the probability that she reads Hindi
newspaper.
(iv) Explain the role of newspapers in the society.
Assume that each child born is equally likely to be a boy or a girl. If a family
Tas two children, what is the conditional probability that both are girls given
that (i) the youngest is a girl (ii) at least one is a girl? Pre-natal sex
determination is a crime. Comment
Consider the experiment of throwing a die, if a multiple of 3 comes up,
throw the die again and if any other number comes, loss a coin. Find the
conditional probability of the event 'The coin shows a tail', given that 'at
least one die shows a 3'.
A fair coin is tossed. If the coin shows head it is tossed again but if it shows
tail then a die is thrown. Find the conditional probability of the event 'the
die shows a number greater than 4' given that 'there is at least one tail.'
MULTIPLICATIVE LAW OF PROBABILITY
A card is drawn from a pack of 52 cards and then a second card is drawn
without replacement. What is the probability that both the cards drawn are
queens.
A bag contains 10 white and 15 black balls. Two balls are drawn in
succession without replacement. What is the probability that first is white
and second is black?
A bag contains 5 white, 7 red and 8 black balls. If four balls are drawn one
by one without replacement, what is the probability that all are white.
An urn contains 5 white and 8 black balls. A set of three balls are drawn two
times successively, such that the balls are not replaced before the second
draw. Find the probability that the first draw gives 3 white balls and second
draw gives 3 black balls.
INDEPENDENT EVENTS
An unbiased die is thrown twice. Let the event be 'even number on the first
throw' and event B be 'even number on the second throw'. Check the
independence of the events and .
If and are two independent events such that and
, find
If not and and are given to be independent
events, find the value of .
Given that the events and are such that and
. Find if the events are (i) mutually exclusive (ii) independent.
If 𝐴 and 𝐵 are two independent events such that 𝑃(𝐴‾ ∩ 𝐵) = and 𝑃(𝐴 ∩ 𝐵‾ ) = , then find 𝑃(𝐴)
and 𝑃(𝐵)
A bag contains 3 red and 5 black balls and a second bag contains 6 red and 4
black balls. A ball is drawn from each bag. Find the probability that (i) both
are red (ii) both are black.
A bag contains 6 white and 9 black balls. Four balls are drawn at random
at a time. Find the probability for the first draw to give four white and
second draw to give four black balls when the (i) balls are replaced (ii)
balls are not replaced.
A problem in Mathematics is given to three students whose chances of ,
and respectively. What is the probability that the problem will be
solved.
A bag contains 5 white, 7 red and 8 black balls. Four balls are drawn one by
one with replacement. What is the probability that atleast one is white?
INDEPENDENT EXPERIMENTS
Sandeep and Salim appear for an interview for two posts. The probability
of Sandeep's selection is and that of Salim's is 1/3. Find the
probability that only one of them will be selected.
A speaks truth in of the cases, while B in of the cases. In what per
cent of cases are they likely to contradict each other in stating the same
fact? In the cases of contradiction do you think, the statement of B will
carry more weight as he speaks truth in more number of cases than ?*
A bag contains 3 red and 5 black balls and second bag contains 6 red and 4
black balls. A ball is drawn from 'ach bag. Find the probability that one is
red and the other is black.
A bag contains 5 red and 3 green balls and a second bag contains 3 red and
5 green balls. One ball is drawn from first bag and two from second bag.
Find the probability that of the three balls drawn, two are red and one is
green.
Two cards are drawn from a pack of 52 cards, one after the other without
replacement. Find the chance that one of these cards is an ace and the
other is a queen of opposite shade.
A bag contains 5 white and 3 black balls. Four balls are successively drawn
out without replacement. What is the probability that they are
alternatively of different colours.
Three groups of children consist of respectively 3 girls and 1 boy, 2 girls
and 2 boys, 1 girl and 3 boys. One child is selected from each group. Find
the chance that three selected children comprise of 1 girl and 2 boys.
A and B throw a coin alternatively till one of them gets a head and wins
the game. Find their respective probabilities of winning.
You have to keep on trying until you succeed. Comment*.
A and B throw a pair of dice alternately. A wins the game if he gets a total
of 7 and wins the game if he gets a total of 10. If starts the game, then
find their respective probabilities of winning.
LAW OF TOTAL PROBABILITY
A bag contains 2 white and 4 black balls and another bag contains 6 white
and 4 black balls. One bag is chosen at random. From the selected bag, one
ball is drawn. Find the probability that the ball drawn is white.
A bag contains 6 red and 8 black balls and another bag contains 8 red and
6 black balls. A ball is drawn from the first bag and without noticing its
colour is put in the second bag. A ball is the drawn from the second bag.
Find the probability that the ball drawn is red in colour.
An urn contains 10 white and 3 black balls, while another urn contains 3
white and 5 black balls. Two balls are drawn from first urn and put into the
second urn and then a ball is drawn from the latter. Find the probability that
it is a white ball?
There are two bags. The first bag contains 5 white and 3 black balls and the
second bag contains 3 white and 5 black balls. Two balls are drawn at
random from the first bag and are put into the second bag, without noting
their colours. Then two balls are drawn from the second bag. Find the
probability that the balls drawn are white and black.
Bag I contains 4 black and 6 red balls and bag II contains 7 black and 3 red
balls. A die is thrown. If 1 or 2 appears on it, then bag I is chosen,
otherwise bag II. If two balls are drawn at random (without replacement)
from the selected bag, find the probability of one of them being red and
another black.
Bag A contains 6 red and 4 black balls and bag B contains 4 red and 6 black
balls. One ball is drawn at random from bag and placed in bag and then
one ball is drawn from bag and placed in bag . Thereupon, one ball is
drawn from bag . Find the probability that the ball drawn is red.
A person has undertaken a construction job. The probabilities are 0.65 that
there will be strike, 0.80 that the construction job will be completed on
time if there is no strike, and 0.32 that the construction job will be
completed on time if there is a strike. Determine the probability that the
construction job will be completed on time. What is your opinion on
strike?"
Bag I contains 2 white and 3 red balls and bag II contains 4 white and 5 red
balls. One ball is drawn at random from one of the bag and found to be red.
Find the probability that it was drawn from bag II.
Three urns contain 6 red, 4 black ; 4 red, 6 black and 5 red, 5 black balls
respectively. One of the urns is selected at random and a ball is drawn from
it. If the ball drawn is red, find the probability that it is drawn from the first
urn.
A bag contains 4 green and 6 white balls. Two balls are drawn one by one
without replacement. If the second ball drawn is white, what is the
probability that the first ball drawn is also white?
The contents of urns I, II, III are as follows:
Urn I: 1 white, 2 black and 3 red balls
Urn II : 2 white, 1 black and 1 red balls
Urn III : 4 white, 5 black and 3 red balls.
One urn is chosen at random and two balls are drawn which happen to be
white and red. What is the probability that they come from urn .
Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5
black balls. One ball is transferred from Bag I to Bag II and then two
balls are drawn at random (without replacement) from Bag II. The balls
so drawn are found to be both red in colour. Find the probability that the
transferred ball is red.
A man is known to speak truth 3 out of 4 times. He throws a die and
reports that it is a six. Find the probability that it is actually a six.
What is the importance of speaking truth ?*
An insurance company insured 2000 scooter drivers, 4000 car drivers and
6000 truck drivers. The probabilities of their accidents are 0.01, 0.03 and
0.15 respectively. One of the insured persons meets with an accident. What
is the probability that he is a scooter driver?
Assume that the chances of a patient having a heart attack is . Assuming
that a meditation and yoga course reduces the risk of heart attack by
and prescription of certain drug reduces its chances by . 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. Interpret the result and state which of the
above stated method is more beneficial for patient.
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.
A card from a pack of 52 cards is lost. From the remaining cards of the pack,
two cards are drawn and are found to be diamonds. Find the probability of the
missing card to be a diamond.
There are three coins. One is a two headed coin (having head on both
faces), another is a biased coin that comes up tails of the times and
the third is an unbiased coin. One of the three coins is chosen at random
and tossed. It shows heads. What is the probability that it is a two headed
coin?
RANDOM VARIABLE
A person plays a game of tossing a coin thrice. For each head he gains Rs.5
and for each tail he loses Rs. 2. Let denote the amount gained or lost by
the person. Show that is a random variable and exhibit it as a function on
the sample space of the experiment.
Find the probability distribution of the number of heads in three tosses of
a coin.
Find the probability distribution of a number of successes in two tosses of
a die, where a success is defined as a number greater than 4. Sketch its
graph also.
A bag contains 3 white and 4 red balls. Three balls are drawn one by one
replacement. Find the probability distribution of the number of red balls.
An urn contains 4 white and 6 red balls. Four balls are drawn at random
from the urn. Find the probability distribution of the number of white balls.
Two bad eggs are mixed accidentally with 10 good ones. Find the
probability distribution of the number of bad eggs in 3 draws at random,
without replacement, from this lot.
Three numbers are selected at random (without replacement) from first six
positive integers. Let denote the largest of the three numbers obtained.
Find the probability distribution of .
Out of a group of 8 highly qualified doctors in a hospital, 6 are very kind and
cooperative with their patients and so are very popular, while the other two
remain reserved. For a health camp, three doctors are selected at random.
Find the probability distribution of the number of popular doctors.
A random variable has the following probability distribution values of

X: 0 1 2 3 4 5 6 7
P(X): 0 k 2k 2k 3k 𝑘 2𝑘 7𝑘 +k

Find (i)
(ii)
(iii)
(iv)
(v)
MEAN AND VARIANCE OF A RANDOM
VARIABLE
Find the mean, variance and standard deviation of the number of heads in
three tosses of a fair coin (or a simultaneous tosses of three coins).
Two unbiased dice are thrown together at random. What is the expected
value of sum of the numbers shown by the two dice?
Two numbers are selected (without replacement from first six positive
integers. Let denote the larger of the two numbers obtained. Find the
probability distribution of . Also find the mean and variane of the
distribution.
A die is tossed thrice. A 'success' is getting 1 or 6 on a toss. Find the mean and
variance of the number of successes.
Two cards are drawn successively with replacement from a well shuffled pack
of 52 cards. Find the mean and standard deviation of the number of kings.
From a lot of 10 items containing 3 defective items, a sample of 4 items is
drawn at random. Let the random variable denote the number of defective
items in the sample. If the sample is drawn without replacement, find the
mean and variance of .
Find the probability distribution of the number of white balls drawn in a
random draw of 3 balls without replacement from a bag containing 4 white
and 6 red balls. Also find the mean and variance of the distribution.
There is a group of 50 people who are patriotic out of which 20 believe in
nonviolence. Two persons are selected at random out of them. Write the
probability distribution for the selected persons who are non-violent. Also
find the mean of the distribution.
Explain the importance of non-violence in patriotism
A die is thrown 6 times. Getting an odd number is a success. What is the
probability of (i) 5 successes (ii) at least 5 successes (iii) atmost 5
successes (iv) no success.
An unbiased coin is tossed 10 times. Find by using binomial distribution, the
probabitity of getting.
(i) exactly 6 heads (ii) atleast 6 heads (iii) atmost 6 heads (iv) atleast 3 heads
A pair of dice is thrown 7 times. If getting a total of 7 is considered a
success, what is the probability of
(i) no success
(ii) 6 successes
(iii) atleast 6 successes (iv) atmost 6 successes?
For 6 trials of an experiment, let be a binomial variate which satisfies the
relation . Find the probability of success.
A box contains 100 tickets each bearing one of the numbers from 1 to 100 . If 5
tickets are drawn successively, with replacement from the box, find the
probability that all the tickets bear numbers divisible by 10 .
An urn contains 5 white, 7 red and 8 black balls. If four balls are drawn one
by one with replacement, what is the probability that
(i) all are white
(ii) only 3 are white
(iii) none is white
The probability that a bulb produced by a factory will fuse after 150 days of
use is 0.05. Find the probability that out of 5 such bulbs
(i) none
(ii) not more than one
(iii) more than one
(iv) at least one will fuse after 150 days of use.
There are 6% defective items in a large bulk of items. Find the probability that a
sample of 8 items will include not more than 1 defective item.
In a hurdle race, a player has to cross 10 hurdles. The probability that he
will clear each hurdle is . What is the probability that he will knock down
fewer than 2 hurdles?
What is the role of sports in our life.*
An unbiased die is thrown again and again until three sixes are obtained.
Find the probability of obtaining third six in the sixth throw of a die.
How many times must a man toss a fair coin, so that the probability of
having at least one head is more than ?
Six dice are thrown 729 times. How many times do you expect at least three
dice to show five or six?
Find the probability distribution of the number of heads when three coins are
tossed.
Find the probability distribution of the number of doublets in 4 throws of a
pair of dice.
5 bad eggs are mixed with 10 good ones. If 3 eggs are drawn one by one with
replacement, then find the probability distribution of the number of good eggs
drawn.
An urn contains 3 white and 6 red balls. Four balls are drawn one by one
with replacement from the urn. Find the probability distribution of the
number of red balls drawn. Also, find mean and variance of the distribution.
PROPERTIES OF BINOMIAL
DISTRIBUTION
Obtain the binomial distribution whose mean is 10 and standard deviation
is 2
A discrete random variable ' ' has mean score equal to 6 and variance equal to
'2'. Assuming that the underlying distribution of is binomial, what is the
probability when .
If the sum of mean and variance of a binomial distribution is 4.8 for 5 trials
Find the distribution.