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Important Formulas -

1. Probability or Chance

Probability or chance is a common term used in day-to-day life. For example, we generally say, 'it may rain today'. This statement has a certain uncertainty.
Probability is quantitative measure of the chance of occurrence of a particular event.

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2 . Experiment

An experiment is an operation which can produce well-defined outcomes.

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3 . Random Experiment:

An experiment in which all possible outcomes are know and the exact output cannot be predicted in advance, is called a random experiment.

Examples

(a) . Tossing of a fair coin
When we toss a coin, the outcome will be either Head (H) or Tail (T)

Total number of outcome if n coins are thrown once or one coin throw n times = 2ⁿ

for example, if a coin toss 3 times then total number of outcome will be 2³ = 8

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(b) . Throwing an unbiased die
Die is a small cube used in games. It has six faces and each of the six faces shows a different number of dots from 1 to 6. Plural of die is dice.

When a die is thrown or rolled, the outcome is the number that appears on its upper face and it is a random integer from one to six, each value being equally likely.

Total number of outcome if n dice are thrown once or one die throw n times = 6ⁿ

for example, if a die toss 2 times then total number of outcome will be 6² = 36

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(c) . Drawing a card from a pack of shuffled cards
A pack or deck of playing cards has 52 cards which are divided into four categories as given below

Spades , Clubs, Hearts, Diamonds

Each of the above mentioned categories has 13 cards, 9 cards numbered from 2 to 10, an Ace, a King, a Queen and a jack

Hearts and Diamonds are red faced cards whereas Spades and Clubs are black faced cards.

Kings, Queens and Jacks are called face cards

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(d) . Taking a ball randomly from a bag containing balls of different colours

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4 . Sample Space:

When we perform an experiment, then the set S of all possible outcomes is called the sample space.

Examples:

In tossing a coin, S = {H, T}

If two coins are tossed, the S = {HH, HT, TH, TT}.

In rolling a dice, we have, S = {1, 2, 3, 4, 5, 6}.

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5 . Event

Any subset of a Sample Space is an event. Events are generally denoted by capital letters A, B , C, D etc.

Examples

(a) . When a coin is tossed, outcome of getting head or tail is an event

(b) . When a die is rolled, outcome of getting 1 or 2 or 3 or 4 or 5 or 6 is an event

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6 . Probability of Occurrence of an Event:

Let S be the sample and let E be an event.

Then, E ⊆ S.

∴   P(E) =

n(E)/n(S)

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7 . Equally Likely Events

Events are said to be equally likely if there is no waitage for a particular event over the other or each and every event has equal chance to occur .

Examples

(a) . When a coin is tossed, Head (H) or Tail is equally likely to occur.

(b) . When a dice is thrown, all the six faces (1, 2, 3, 4, 5, 6) are equally likely to occur.

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8 .Mutually Exclusive Events

Two or more than two events are said to be mutually exclusive if the occurrence of one of the events excludes the occurrence of the other

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This can be better illustrated with the following examples

(a) . When a coin is tossed, we get either Head or Tail. Head and Tail cannot come simultaneously. Hence occurrence of Head and Tail are mutually exclusive events.

(b) . When a die is rolled, we get 1 or 2 or 3 or 4 or 5 or 6. All these faces cannot come simultaneously. Hence occurrences of particular faces when rolling a die are mutually exclusive events.

Let throw a die once and there are two event A and B , A denote even numbers B denotes odd numbers on the upper face of die .
So A = {2,4,6}, B = {1,3,5} There is no common element between A and B so A and B are know as mutually exclusive event .

Note : If A and B are mutually exclusive events, A ∩ B = ∅ where ∅ represents empty set.

(c) . Consider a die is thrown and A be the event of getting 2 or 4 or 6 and B be the event of getting 4 or 5 or 6. Then

A = {2, 4, 6} and B = {3, 5, 6}

Here A ∩ B ≠ ∅ , Hence A and B are not mutually exclusive events.

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9 . Independent Events

Events can be said to be independent if the occurrence or non-occurrence of one event does not influence the occurrence or non-occurrence of the other.

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10 . Simple and Compound Events

In the case of simple events, we take the probability of occurrence of single events.

Examples

(a) . Probability of getting a Tail (T) when a coin is tossed

(b) . Probability of getting 4 when a die is thrown

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In the case of compound events, we take the probability of joint occurrence of two or more events.

Examples

(a) . When two coins are tossed, probability of getting a Head (H) in the first toss and getting a Tail (T) in the second toss.

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11 .Exhaustive Event

Exhaustive Event is the total number of all possible outcomes of an experiment.

Examples

(a) . When a coin is tossed, we get either Head or Tail. Hence there are 2 exhaustive events.

(b) . When two coins are tossed, the possible outcomes are (H, H), (H, T), (T, H), (T, T). Hence there are 4 exhaustive events.
Note :   Let there are two event A and B , and A ∪ B = S(sample space) then event A and B are known as exhaustive event .

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11 .Algebra of Events

Let A and B are two events with sample space S. Then

(a) . A ∪ B is the event that either A or B or Both occur. (i.e., at least one of A or B occurs)

(b) . A ∩ B is the event that both A and B occur

(c) . A̅ is the event that A does not occur

(d) . A̅ ∩ B̅ is the event that none of A and B occurs

(e) . A̅ ∩ B is the event that A does not occur but B occur .

(f) . A ∩ B̅ is the event that A occur but B does not occur .

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12 . Conditional Probability

Let A and B be two events associated with a random experiment. Then, probability of the occurrence of A given that B has already occurred is called conditional probability and denoted by P(A/B)

Example : A bag contains 5 red and 4 blue balls. Two balls are drawn from the bag one by one without replacement. What is the probability of drawing a blue ball in the second draw if a red ball is already drawn in the first draw?

Let A be the event of drawing red ball in the first draw and B be the event of drawing a blue ball in the second draw. Then, P(B/A) = Probability of drawing a blue ball in the second draw given that a red ball is already drawn in the first draw.

Total Balls = 5 + 4 = 9

Since a red ball is drawn already,

total number of balls left after the first draw = 8

total number of blue balls after the first draw = 4

P(B/A) =

4/8

=

1/2

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13 . Results on Probability:

(a) .   P(S) = 1 (probability of sample space)

(b) .    0 ≤ P (E) ≤ 1

(c) .   Probalility of impossible event is = 0 and probability of sure event is 1.

(d) .   For any events A and B we have : P(A ∪ B) = P(A) + P(B) - P(A ∩ B)(addition theorem, it means occurence of either A or B or both)

(e) .   If A̅ denotes (not-A), then P(A) = 1 - P(A̅).

(f) .    P(A ∩B) = P(A).P(B)(this condition follows if A and B are independent event)

Note: If A and B are independent event then

(i) A' and B are independent event.

(ii)A and B' are independent event.

(iii) A' and B' are independent event.

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