In life, things are not always certain. Consider the following situations:

- A candidate appearing for an interview for a job may or may not get the job
- It may or may not snow today.
- If a coin is tossed you might get a head or a tail (or neither if the coin falls on its edge).
- When a dice is thrown your chance of getting a 6 may or may not occur, since it is equally likely that a 1, 2, 3, 4, or 5 may turn up.

These situations have no definite answer. They involve an element of uncertainty. The 'Probability Theory' is designed to estimate the degree of uncertainty regarding the happening of a given phenomenon.

Probability is used in various situations in physical, biological, and social sciences.

Suppose you toss a dice once, what are the possible outcomes? A dice, obviously can fall with any of its faces uppermost. The number on each face is a possible **outcome**. If the dice is well-balanced, it is likely to show a 2, or a 1, 3, 4, 5, 6.

Since there are 6 equally likely outcomes 1, 2, 3, 4, 5 or 6 in a single throw of a dice and there is only one way of getting a particular outcome say ‘ 6 ', therefore the chance of getting 6 is one in six. Or the probability of getting a 6 is 1/6. We write this as P (6) = 1/6

Similarly if you toss a coin it can show a head (H) or a tail (T). So there are only two equally likely outcomes. The probability of getting a tail is one in two or 1/2.

The outcome is also called an event (E). We write the probability of an event as P (E) and define it as

In the case of tossing a dice, the total number of outcomes in the set {1, 2, 3, 4, 5, 6} is 6. If we want a 6 then we just have 1 favorable outcome as there's only one outcome of 6 on a dice.

The set {1, 2, 3, 4, 5, 6} is called a **Sample space** and each **Outcome** is called a **Sample point**. Tossing a coin or a dice is called a **Random**

What is the probability of getting an '8' if a dice is tossed once? Since none of the faces is marked by an 8, getting an eight is **impossible**. Such an event is called an **impossible event** and we have

**The Probability of an impossible event is zero.**

The probability of getting a 1, 2, 3, 4, 5 or 6 when a dice is tossed i.e. a number less than 7 is certain to happen or P (< 7) = 6/6 = 1.

This is called a **sure event**.

We know that the probability of getting a number 6 in the throw of a dice is 1/6. What is the probability of getting a number other than 6? The numbers are 1, 2, 3, 4 or 5 or 5 favorable outcomes.

We can write P (other than 6) as P (not 6)

Then

We thus have the formula for any event E

P (E) + P ( Ē ) = 1

and P (E) = 1-P( Ē )

P (Ē) = 1- P (E)

P ( Ē) indicates P(not E).

A dice is thrown once. What is the probability of getting

- A number 3 or 4?
- An odd number?
- A prime number?

**Solution:**

- Number of favorable outcomes getting a 3 or 4 = 2

Total number of outcomes = 6

Required probability: P(E)

- Number of odd numbers = 3, since the only odd numbers are 1, 3, and 5

Number of favorable outcomes = 3

Total number of outcomes = 6

Probability of event = P(E)

- Number of prime number = 3, since the only prime numbers are (2, 3, 5)

Number of favorable outcomes = 3

Total number of outcomes = 6

Probability of the event = P(E)

If one card is drawn from a well shuffled deck of 52 cards, find the probability that the card is

- a diamond
- an ace
- a black card
- not a diamond
- not an ace
- not a black card
- a club or a heart
- a club and a king

**Solution:**

- In a pack of 52 cards there are 13 diamond cards

Number of favorable outcomes = 13.

Total number of outcomes = 52

Probability of the event = P(E)

(getting diamond) - Number of aces in a deck of 52 cards is 4

Number of favorable outcomes = 4

Total number of outcomes = 52

Probability of the event = P(E)

(getting an ace)

- The number of black cards in a deck of 52 cards is 26

13 of clubs

13 of spades

Number of favorable outcomes = 26

Total number of outcomes = 52

Probability of the event = P(E)

(getting a black card)

- Since the probability P (E) of getting a diamond is 1/4

The probability of not getting a diamond

- Probability of getting an ace = P (E) = 1/13

Probability of not getting an ace = 1-P (E)

- Probability of getting a black card = P (E) = 1/2

Probability of not getting a black card = P (Ē) = 1 – P (E)

- Probability of getting a club or a heart

Number of club cards in a deck = 13

Number of heart cards in a deck = 13

Number of favorable outcomes = 13 + 13 = 26 Total number of outcomes = 52

Probability of the event = P(E)

(getting a club or a heart)

- A club and a king

Since there is only one king of clubs

Number of favorable outcomes = 1

Total number of outcomes = 52

Probability of the event = P(E)

(getting a club or a king)

- A dice is thrown twice. Find the probability of getting a sum of six in these throws.
- A coin is tossed thrice, write its sample space.
- From a well shuffled deck of cards find the probability of
- Getting '2' of hearts.
- Getting a king or a queen or a jack.
- Not getting an ace.
- Getting a red card

**Answer:**

- A dice is thrown twice.

Sample space S = {(1,1), (1, 2),(1,3), (1,4),(1,5), (1,6)

(2,1), (2, 2),(2,3), (2, 4),(2,5), (2, 6)

(3,1), (3, 2),(3,3), (3, 4),(3,5), (3, 6)

(4,1), (4, 2),(4,3), (4, 4),(4,5), (4, 6)

(5,1), (5, 2),(5,3), (5, 4),(5,5), (5, 6)

(6,1), (6, 2),(6,3), (6, 4),(6,5), (6, 6)}

Total number of outcomes= 36 = n(S)

A = event of getting a sum of six

> - A coin is tossed thrice Sample space S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT} Total number of outcomes = 23 = 2 * 2 * 2 = 8
- Total number of outcomes = Total number of cards
= n(S)
n(S) = 52
- A = event of getting ' 2 ' of hearts n(A) =1 = number of favorable outcomes Probability of getting a ' 2 ' of hearts is
- A = event of getting a king

n(A) = number of favorable outcomes = 4

B = event of getting a queen

n(B) = number of favorable outcomes = 4

C = event of getting a jack

n(c) = number of favorable outcomes = 4

Since getting a king or a queen or a jack are mutually exclusive events

P(A or B or C) = P (A ∪ B ∪ C) = P(A) + P (B) + P (C)

- A = event of getting an ace

n(A) = number of favorable outcomes = 4

- R = event of getting a red card

n(R) = number of favorable outcomes

= 26 (13 diamonds + 13 hearts)

s Probability of getting a red card is

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