Download Probability Topics

Probability
Topics
a. Experiment: A process of obtaining definite outcomes or results.
b. Sample space: It a set of all possible outcomes or results.
c. Event: It a subset of a sample space consisting of some or all elements of sample space.
S.# Experiments
Sample Space
A few Events
1
Tossing a coin
{ head, tail}
{Head}, {Tail} etc.
2
Throwing a dice
{1, 2, 3, 4, 5, 6}
{1,2,6}, {3,6}, {2, 3, 5, 6}
3
Picking up a ball from a bag {Red, Blue, Green} { R, B}, {R, G, B} etc
d. Number of elements of a set: The number of elements in a set A is denoted by 𝑛(𝐴).
 If A = {a, b, c, d} then 𝑛(𝐴) = 4
 If B = {1, 2, 3, 4, 5, 6,11} then 𝑛(𝐵) = 7
e. Probability: The probability of an event E of a sample space is denoted by 𝑃(𝐸) and is given by
𝑃(𝐸) =
f.
𝑛(𝐸)
𝑛(𝑆)
Set and Operation in Sets:
 Intersection of Two sets: Intersection of two sets A and B is denoted by 𝑨 ∩ 𝑩 and it is set of
common elements of set 𝐴 and Set 𝐵.
 Union of Two sets: Union of two sets A and B is denoted by 𝑨 ∪ 𝑩 and it asset of common and
non common elements of set 𝐴 and Set 𝐵.
 Difference of two sets: Difference of two sets A an dB is denoted by 𝑨 − 𝑩 and it is a set of
elements of A which are not in B and 𝑩 − 𝑨 is a set of elements of B which are not in B Usually
𝑩−𝑨≠𝑨−𝑩
 Complement of a set: Complement of a event 𝐴 of a sample space 𝑆 is denoted by 𝐴𝑐 or 𝐴/ and
it is given by 𝐴𝑐 = 𝑆 − 𝐴
Venn diagram: It is used to represent the Set Operation i.e. Union, Intersection and complement
g.
of sets. In Venn diagram the Sample space is usually represented by a rectangle and sets are
represented by circles. Union is Addition of Areas of components, Intersection is Common areas
of components; Complement is Area of other than component
h. Some common Venn diagrams..
A
A
A⋂B
A⋂𝐵𝑐
B
B
A
A
B⋂𝐴𝑐
AUB
B
B
A
B
𝑐
𝐴
A
i.
Some Basics Laws of probability:
 P(A) + P(Ac ) = 1
 0 ≤ P(E) ≤ 1
 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
 𝑃(𝐵) = 𝑃(𝐴 ∩ 𝐵𝑐 ) + 𝑃(𝐴 ∩ 𝐵)
 𝑃(𝐴𝑐 ∩ 𝐵𝑐 ) = 𝑃(𝐴 ∪ 𝐵)
 Probability of Event A or B or both means 𝑃(𝐴 ∪ 𝐵)
 Probability of Event Either A or B means 𝑃(𝐴 ∪ 𝐵)
 Probability of Event A and B means 𝑃(𝐴 ∩ 𝐵)
 Probability of Event A but not B means 𝑃(𝐴 ∩ 𝐵𝑐 )
 Probability of Event neither A nor B means 𝑃(𝐴𝑐 ∩ 𝐵𝑐 ).
Q.1
An ordinary die is thrown. Find the probability that the number obtained is
i. A multiple of 3,
iii. A factor of 6.
ii. Less than 7,
Q.2
In a box of highlighter there are eight which have dried up and will not write. The box
contains 10 red, 15 blue, 5 green and 10 yellow highlighters. A highlighter is picked up at
random from the box. Find the probability that
i. It is blue,
iv. It is purple,
ii. It is neither green nor yellow,
v. It will write.
iii. It is not yellow,
Q.3
The probability of an event occurring is 0.27. Find the probability that it will not occur.
j. Possibility Diagram: It is used to represent the results of two experiments usually when the
results are in the forms of letters or numeric. For example when two coins are tossed or a coin
is tossed twice. A coin is tossed and a dice is thrown.
Q.4
Two ordinary unbiased dice are thrown. Find the probability that
i. The sum of two dice is 3,
ii. The sum of the two dice exceeds 9,
iii. The two dice show the same results,
iv. The numbers on the two dice differ by more than 2.
Q.5 In a group of 30 students all study at least one of the subjects Physics and Biology, 20
attend the Physics class and 21 attend the Biology class. Find the probability that a student
chosen at random studies both Physics and Biology.
2
1
1
Q.6 Given that 𝑃(𝐴𝑐 ) = , 𝑃(𝐵) = and 𝑃(𝐴 ∩ 𝐵) =
find 𝑃(𝐴 ∪ 𝐵).
3
2
12
k. Exclusive or Mutually exclusive Events: If A and B are two events of a same sample space then
they are said to be mutually exclusive if their intersection is empty set .
If A and B are mutually exclusive then 𝑃(𝐴 ∩ 𝐵) = 0
l. Exhaustive Events: If A and B are two events are such that between them they make up the
whole sample space. If A and B are Exhaustive then 𝑃(𝐴 ∪ 𝐵) = 1
m. Conditional probability: If A and B are two events, not necessarily from the same experiment,
then conditional probability that A occurs, given that B has already occurs is written as
𝑃(𝐴|𝐵) and given by
𝑃(𝐴|𝐵) =
𝑛(𝐴∩𝐵)
𝑛(𝐵)
Or 𝑃(𝐴|𝐵) =
𝑃(𝐴∩𝐵)
𝑃(𝐵)
𝑃(𝐴|𝐵) × 𝑃(𝐵) = 𝑃(𝐵|𝐴) × 𝑃(𝐴)
When a dice is thrown the score was an odd number. What is the probability that it was
a prime number
Q.8
In a group of 100 people, 40 own a cat, 25 own a dog and 15 own a cat and a dog. Find
the probability that a person chosen at random
i. Own a cat or a dog,
ii. Owns a dog or a cat, but not both,
iii. Own a dog, given that he owns a cat,
iv. Does not own a cat, given that he owns a dog.
1
Q.9
The probability that a person in a particular evening class is left handed is 6. From a class
of 15 women and 5 men a person is chosen at random , find the probability that the
person chosen is a man or is left handed.
n. Probability Tree diagram: It is used to represent the results of two or more than two
experiments with their probabilities.
Basics Rules:
i. It is started from a single point called the vertex.
ii. Branches from the vertex are made equal to the number of possible outcomes of first
experiment with their probability.
iii. From each branch of the first experiment, branches are drawn equal to the possible
outcomes of the second experiment and similarly tree is formed for third fourth and
fifth experiments.
iv. Number of columns is equal to the number of experiments.
v. Product of the probabilities from vertex to the end branch gives the probability of that
branch.
vi. If some probabilities are collected from last columns then the results are added.
vii. The sum of probabilities of a vertex or semi vertex is always equal to 1.
viii. The sum of probabilities of last column is 1.
ix. Usually probability diagrams are of two types, one is with replacement and other is
without replacement.
Q.10 The probability that I am late from work is 0.05. Find the probability that on two
consecutive mornings,
i. I am late for work twice,
ii. I am late for work once.
Q.11 In a restaurant 40% of the customers choose steak for their main course. If a customer
chooses a steak, the probability that he will choose Ice cream is 0.6. If he does not have
steak , the probability that he will choose ice cream is 0.3. Find the probability that a
customer picked at random will choose
i. Steak and ice cream,
ii. Ice cream.
Q.12 A box contains 20 chocolates, of which 15 have soft centers and 5 had hard centers. Two
chocolates are taken at random, one after the other. Calculate the probability that
i. Both chocolates have soft centers,
ii. One of each sort of chocolate is taken,
iii. Both chocolate have hard centers , given that the second chocolate had a hard center.
Also
Q.7
Answers :
2
Q.1
1
1
3
Q.2
0.375 0.625 0.75
0.8
Q.3
0.73
1
1
1
Q.4
18
6
6
Q.5
11
30
Past Papers Questions
Q.1
Q.2
Q.3
Q.6
0
1
3
Q.7
Q.8
Q.9
Q.10
Q.11
Q.12
3
4
2
3
0.5
0.35
0.375 0.4
3
8
0.0025 0.095
0.24 0.42
21
38
15
38
20
83