Probability | Probability Formula | Probability of an Event | Equally Likely Events | Complementary Events | Solved Examples, FAQs

📘 Introduction to Probability – Class 11 Mathematics
Written by Neeraj Anand
Published by ANAND TECHNICAL PUBLISHERS

🔍 What is Probability?

Probability is a branch of mathematics that deals with uncertainty. It measures the likelihood or chance of an event occurring. The value of probability always lies between 0 and 1, where:

• 0 indicates an impossible event.
• 1 indicates a certain event.

In everyday life, probability helps us predict outcomes, from weather forecasts to predicting chances of winning in games.

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🎯 Basic Terminology in Probability

1. Experiment:
   An action or process that leads to well-defined outcomes.
   Example: Tossing a coin or rolling a die.
2. Sample Space (S):
   The set of all possible outcomes of an experiment.
   Example: For a die roll, S={1,2,3,4,5,6}
3. Event:
   A subset of the sample space. It could be a single outcome or a group of outcomes.
   Example: Getting an even number when rolling a die: E={2,4,6}
4. Equally Likely Events:
   Events that have the same chance of occurring.
   Example: In a fair coin toss, getting heads or tails are equally likely events.

What is Probability ?

Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one.

For example, when we toss a coin, either we get Head OR Tail, only two possible outcomes are possible (H, T). But when two coins are tossed then there will be four possible outcomes,  i.e {(H, H), (H, T), (T, H), (T, T)}.

🔑 Types of Probability

1. Theoretical Probability: Based on reasoning or calculations without actual experiments.
2. Experimental Probability: Based on actual experiments and observations.
3. Subjective Probability: Based on intuition, beliefs, or personal judgment.

Formula for Probability

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.

Probability of event to happen P(E) = Number of favourable outcomes/Total Number of outcomes

📝 Real-Life Examples

• Tossing a coin and predicting heads or tails.
• Drawing a card from a deck and predicting its suit.
• Predicting rainfall on a particular day.

Solved Examples

1) There are 6 pillows in a bed, 3 are red, 2 are yellow and 1 is blue. What is the probability of picking a yellow pillow?

Ans: The probability is equal to the number of yellow pillows in the bed divided by the total number of pillows, i.e. 2/6 = 1/3.

2) There is a container full of coloured bottles, red, blue, green and orange. Some of the bottles are picked out and displaced. Amit did this 1000 times and got the following results:

• No. of blue bottles picked out: 300
• No. of red bottles: 200
• No. of green bottles: 450
• No. of orange bottles: 50

a) What is the probability that Amit will pick a green bottle?

Ans: For every 1000 bottles picked out, 450 are green.

Therefore, P(green) = 450/1000 = 0.45

b) If there are 100 bottles in the container, how many of them are likely to be green?

Ans: The experiment implies that 450 out of 1000 bottles are green.

Therefore, out of 100 bottles, 45 are green.

Probability of an Event

Assume an event E can occur in r ways out of a sum of n probable or possible equally likely ways. Then the probability of happening of the event or its success is expressed as;

P(E) = r/n

The probability that the event will not occur or known as its failure is expressed as:

P(E’) = (n-r)/n = 1-(r/n)

E’ represents that the event will not occur.

Therefore, now we can say;

P(E) + P(E’) = 1

This means that the total of all the probabilities in any random test or experiment is equal to 1.

For example, if you want to find the probability of rolling a 4 on a six-sided die, there is 1 favorable outcome (rolling a 4) out of 6 possible outcomes (1, 2, 3, 4, 5, 6). Therefore,

P(rolling a 4)= 1/6

What are Equally Likely Events?

When the events have the same theoretical probability of happening, then they are called equally likely events.

The results of a sample space are called equally likely if all of them have the same probability of occurring.

For example, if you throw a die, then the probability of getting 1 is 1/6.

Similarly, the probability of getting all the numbers from 2,3,4,5 and 6, one at a time is 1/6. Hence, the following are some examples of equally likely events when throwing a die:

• Getting 3 and 5 on throwing a die
• Getting an even number and an odd number on a die
• Getting 1, 2 or 3 on rolling a die

are equally likely events, since the probabilities of each event are equal.

Complementary Events

The possibility that there will be only two outcomes which states that an event will occur or not. Like a person will come or not come to your house, getting a job or not getting a job, etc. are examples of complementary events.

Basically, the complement of an event occurring in the exact opposite that the probability of it is not occurring. Some more examples are:

• It will rain or not rain today
• The student will pass the exam or not pass.
• You win the lottery or you don’t.

🧠 Important Properties of Probability

• The probability of an event lies between 0 and 1.
  0 ≤ P(E) ≤ 1
• The sum of probabilities of all outcomes in a sample space is 1.
  P(S)=1
• The probability of an impossible event is 0.
• The probability of a certain event is 1.

Solved Examples on Probability Class 11 Math

Question 1: Find the probability of ‘getting 3 on rolling a die’.

Solution:

Sample Space = S = {1, 2, 3, 4, 5, 6}

Total number of outcomes = n(S) = 6

Let A be the event of getting 3.

Number of favourable outcomes = n(A) = 1

i.e. A  = {3}

Probability, P(A) = n(A)/n(S) = 1/6

Hence, P(getting 3 on rolling a die) = 1/6

Question 2: Draw a random card from a pack of cards. What is the probability that the card drawn is a face card?

Solution:

A standard deck has 52 cards.

Total number of outcomes = n(S) = 52

Let E be the event of drawing a face card.

Number of favourable events = n(E) = 4 x 3 = 12 (considered Jack, Queen and King only)

Probability, P = Number of Favourable Outcomes/Total Number of Outcomes

P(E) = n(E)/n(S)

= 12/52

= 3/13

P(the card drawn is a face card) = 3/13

Question 3: A vessel contains 4 blue balls, 5 red balls and 11 white balls. If three balls are drawn from the vessel at random, what is the probability that the first ball is red, the second ball is blue, and the third ball is white?

Solution:

Given,

The probability to get the first ball is red or the first event is 5/20.

Since we have drawn a ball for the first event to occur, then the number of possibilities left for the second event to occur is 20 – 1 = 19.

Hence, the probability of getting the second ball as blue or the second event is 4/19.

Again with the first and second event occurring, the number of possibilities left for the third event to occur is 19 – 1 = 18.

And the probability of the third ball is white or the third event is 11/18.

Therefore, the probability is 5/20 x 4/19 x 11/18 = 44/1368 = 0.032.

Or we can express it as: P = 3.2%.

Question 4: Two dice are rolled, find the probability that the sum is:

1. equal to 1
2. equal to 4
3. less than 13

Solution:

To find the probability that the sum is equal to 1 we have to first determine the sample space S of two dice as shown below.

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) }

So, n(S) = 36

1) Let E be the event “sum equal to 1”. Since, there are no outcomes which where a sum is equal to 1, hence,

P(E) = n(E) / n(S) = 0 / 36 = 0

2) Let A be the event of getting the sum of numbers on dice equal to 4.

Three possible outcomes give a sum equal to 4 they are:

A = {(1,3),(2,2),(3,1)}

n(A) = 3

Hence, P(A) = n(A) / n(S) = 3 / 36 = 1 / 12

3) Let B be the event of getting the sum of numbers on dice is less than 13.

From the sample space, we can see all possible outcomes for the event B, which gives a sum less than B. Like:

(1,1) or (1,6) or (2,6) or (6,6).

So you can see the limit of an event to occur is when both dies have number 6, i.e. (6,6).

Thus, n(B) = 36

Hence,

P(B) = n(B) / n(S) = 36 / 36 = 1

Frequently Asked Questions(FAQs) on Probability Class 11 Math

Q1

What is a random experiment in probability?

A random experiment is an experiment in probability, with more than one possible outcome and for which we cannot predict the outcome in advance.

Q2

What do you learn about sample space in probability class 11?

Sample space is the set of all the possible outcomes.

Q3

What are the types of events taught in Class 11 probability?

The types of events are Impossible and Sure Events, Simple Events and Compound Events.

Q4

What is probability? Give an example

Probability is a branch of mathematics that deals with the occurrence of a random event. For example, when a coin is tossed in the air, the possible outcomes are Head and Tail.

Q5

What is the formula of probability?

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes to the total number of outcomes.

Q6

What are the different types of probability?

There are three major types of probabilities:
Theoretical Probability
Experimental Probability
Axiomatic Probability

Q7

What are the basic rules of probability?

If A and B are two events, then;
P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B )
P ( A ∩ B ) = P ( B ) ⋅ P ( A | B )

Q8

What is the complement rule in probability?

In probability, the complement rule states that “the sum of probabilities of an event and its complement should be equal to 1”. If A is an event, then the complement rule is given as:
P(A) + P(A’) = 1.

Q9

What are the different ways to present the probability value?

The three ways to present the probability values are:

• Percentage
• Ratio
• Decimal or fraction

Q10

What does the probability of 0 represent?

The probability of 0 represents that the event will not happen or that it is an impossible event.

Q11

What is the sample space for tossing two coins?

The sample space for tossing two coins is:
S = {HH, HT, TH, TT}

Probability Problems

1. Two dice are thrown together. Find the probability that the product of the numbers on the top of the dice is:
   (i) 6 (ii) 12 (iii) 7
2. A bag contains 10 red, 5 blue and 7 green balls. A ball is drawn at random. Find the probability of this ball being a
   (i) red ball (ii) green ball (iii) not a blue ball
3. All the jacks, queens and kings are removed from a deck of 52 playing cards. The remaining cards are well shuffled and then one card is drawn at random. Giving ace a value 1 similar value for other cards, find the probability that the card has a value
   (i) 7 (ii) greater than 7 (iii) less than 7
4. A die has its six faces marked 0, 1, 1, 1, 6, 6. Two such dice are thrown together and the total score is recorded.
   (i) How many different scores are possible?
   (ii) What is the probability of getting a total of 7?

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