Class 10 Math Chapter – 14 Probability CBSE Board

Probability

Probability

1. Probability is a quantitative measure ofuncertainty.
2. In the experimental approach to probability, we find the probability of the occurrence of an eventby actually performing the experiment a number of times and adequate recording of the happening of event.
3. In the theoretical approach to probability, we try to predict what will happen without actually performing theexperiment.
4. The experimental probability of an event approaches to its theoretical probability if the number oftrials of an experiment is very large.
5. An outcome is a result of a single trial of anexperiment.
6. The word ‘experiment’ means an operation which can produce some well definedoutcome(s). There are two types ofexperiments:
   1. Deterministic experiments: Experiments which are repeated under identical conditionsproduce the same results or outcomes are called deterministicexperiments.
   1. Random or Probabilistic experiment: If an experiment, when repeated under identical conditions, do not produce the same outcome every time but the outcome in a trial is one of the severalpossible outcomes, then it is known as a random or probabilisticexperiment.

In this chapter, the term experiment will stand for random experiment.

• The collection of all possible outcomes is called the samplespace.
• An outcome of a random experiment is called an elementaryevent.
• An event associated to a random experiment is a compound event if it is obtained by combiningtwo or more elementary events associated to the randomexperiment.
• An event A associated to a random experiment is said to occur if any one of the elementaryevents associated to the event A is anoutcome.
• An elementary event is said to be favorable to a compound event A, if it satisfies the definition of the compound event A. In other words, an elementary event E is favorable to a compound event A, if we say that the event A occurs when E is an outcome of atrial.
• In an experiment, if two or more events have equal chances to occur or have equal probabilities,then they are called equally likely events.

The theoretical probability (also called classical probability) of an event E, written as P (E), is definedas

1. For two events A and B of anexperiment:

If P(A) >P(B) then event A is more likely to occur than event B.

If P(A) = P(B) then events A and B are equally likely to occur.

• An event is said to be sure event if it always occur whenever the experiment is performed. The probability of sure event is always one. In case of sure event elements are same as the sample space.
• An event is said to be impossible event if it never occur whenever the experiment is performed.The probability of an impossible event is always zero. Also, the number of favorable outcome is zero for an impossibleevent.
• Probability of an event lies between 0 and 1, both inclusive, i.e., 0 ≤P (A) ≤1
• If Eisanevent inarandom experiment thentheevent‘notE’(denoted byE)is called the complementary event corresponding to E.
• The sum of the probabilities of all elementary events of an experiment is1.
• For an event E, P() =1 –P(E) , where the event representing ‘not E” is thecomplement of eventE.

8.  Suits of Playing Card

A pack of playing cards consist of 52 cards which are divided into 4 suits of 13 cards each. Each suit consists of one ace, one king, one queen, one jack and 9 other cards numbered from 2 to 10. Four suits are named as spades, hearts, diamonds and clubs.

1. FaceCards

King, queen and jack are face cards.

The formula for finding the geometric probability of an event is given by:

Here, ‘measure’ may denote length, area or volume of the region or space.

Event and outcome

An Outcome is a result of a random experiment. For example, when we roll a dice getting six is an outcome.

An Event is a set of outcomes. For example when we roll dice the probability of getting a number less than five is an event.

Note: An Event can have a single outcome

Events and Types of Events in Probability

What are Events in Probability?

A probability event can be defined as a set of outcomes of an experiment. In other words, an event in probability is the subset of the respective sample space. So, what is sample space?

The entire possible set of outcomes of a random experiment is the sample space or the individual space of that experiment. The likelihood of occurrence of an event is known as probability. The probability of occurrence of any event lies between 0 and 1.

The sample space for the tossing of three coins simultaneously is given by:

S = {(T , T , T) , (T , T , H) , (T , H , T) , (T , H , H ) , (H , T , T ) , (H , T , H) , (H , H, T) ,(H , H , H)}

Suppose, if we want to find only the outcomes which have at least two heads; then the set of all such possibilities can be given as:

E = { (H , T , H) , (H , H ,T) , (H , H ,H) , (T , H , H)}

Thus, an event is a subset of the sample space, i.e., E is a subset of S.

There could be a lot of events associated with a given sample space. For any event to occur, the outcome of the experiment must be an element of the set of event E.

What is the Probability of Occurrence of an Event?

The number of favourable outcomes to the total number of outcomes is defined as the probability of occurrence of any event. So, the probability that an event will occur is given as:

Types of Events in Probability:

Some of the important probability events are:

• Impossible and Sure Events
• Simple Events
• Compound Events
• Independent and Dependent Events
• Mutually Exclusive Events
• Exhaustive Events
• Complementary Events
• Events Associated with “OR”
• Events Associated with “AND”
• Event E1 but not E2

Impossible and Sure Events

If the probability of occurrence of an event is 0, such an event is called an impossible event and if the probability of occurrence of an event is 1, it is called a sure event. In other words, the empty set ϕ is an impossible event and the sample space S is a sure event.

Simple Events

Any event consisting of a single point of the sample space is known as a simple event in probability. For example, if S = {56, 78 , 96 , 54 , 89} and E = {78} then E is a simple event.

Compound Events

Contrary to the simple event, if any event consists of more than one single point of the sample space then such an event is called a compound event. Considering the same example again, if S = {56 ,78 ,96 ,54 ,89}, E1 = {56 ,54 }, E2 = {78 ,56 ,89 } then, E₁ and E₂ represent two compound events.

Independent Events and Dependent Events

If the occurrence of any event is completely unaffected by the occurrence of any other event, such events are known as an independent event in probability and the events which are affected by other events are known as dependent events.

Mutually Exclusive Events

If the occurrence of one event excludes the occurrence of another event, such events are mutually exclusive events i.e. two events don’t have any common point. For example, if S = {1, 2 , 3 , 4 , 5 , 6} and E1, E2 are two events such that E1 consists of numbers less than 3 and E2 consists of numbers greater than 4.

So, E1 = {1,2} and E2 = {5,6} .

Then, E1 and E2 are mutually exclusive.

Exhaustive Events

A set of events is called exhaustive if all the events together consume the entire sample space.

Complementary Events

For any event E1 there exists another event E1‘ which represents the remaining elements of the sample space S.

E1 = S − E1‘

If a dice is rolled then the sample space S is given as S = {1 , 2 , 3 , 4 , 5 , 6 }. If event E1 represents all the outcomes which is greater than 4, then E1 = {5, 6} and E1‘ = {1, 2, 3, 4}.

Thus E1‘ is the complement of the event E1.

Similarly, the complement of E₁, E₂, E₃……….E_n will be represented as E₁‘, E₂‘, E₃‘……….E_n‘

Experimental Probability

Experimental probability can be applied to any event associated with an experiment that is repeated a large number of times.

A trial is when the experiment is performed once. It is also known as empirical probability.

Experimental or empirical probability: P(E) =Number of trials  where the event occurred/Total Number of Trials

You and your 3 friends are playing a board game. It’s your turn to roll the die and to win the game you need a 5 on the dice. Now, is it possible that upon rolling the die you will get an exact 5? No, it is a matter of chance. We face multiple situations in real life where we have to take a chance or risk. Based on certain conditions, the chance of occurrence of a certain event can be easily predicted. In our day to day life, we are more familiar with the word ‘chance and probability’. In simple words, the chance of occurrence of a particular event is what we study in probability. In this article, we are going to discuss one of the types of probability called  “Experimental Probability” in detail.

Theoretical Probability

Theoretical Probability, P(E) = Number of Outcomes Favourable to E / Number of all possible outcomes of the experiment

Here we assume that the outcomes of the experiment are equally likely.

Every one of us would have encountered multiple situations in life where we had to take a chance or risk. Depending on the situation, it can be predicted up to a certain extent if a particular event is going to take place or not. This chance of occurrence of a particular event is what we study in probability. In our everyday life, we are more accustomed to the word ‘chance’ as compared to the word ‘probability’. Since Mathematics is all about quantifying things, the theory of probability basically quantifies these chances of occurrence or non-occurrence of certain events. In this article, we are going to discuss what is probability and its two different types of approaches with examples.

In Mathematics, the probability is a branch that deals with the likelihood of the occurrences of the given event. The probability value is expressed between the range of numbers from 0 to 1. The three basic rules connected with the probability are addition, multiplication, and complement rules.

Theoretical Probability Vs Experimental Probability

Probability theory can be studied using two different approaches:

Theoretical Probability

Experimental Probability

Theoretical Probability Definition

Theoretical probability is the theory behind probability. To find the probability of an event using theoretical probability, it is not required to conduct an experiment. Instead of that, we should know about the situation to find the probability of an event occurring. The theoretical probability is defined as the ratio of the number of favourable outcomes to the number of possible outcomes.

Probability of Event P(E) = No. of. Favourable outcomes/ No. of. Possible outcomes.

Experimental Probability Definition

The experimental probability also is known as an empirical probability, is an approach that relies upon actual experiments and adequate recordings of occurrence of certain events while the theoretical probability attempts to predict what will happen based upon the total number of outcomes possible. The experimental Probability is defined as the ratio of the number of times that event occurs to the total number of trials.

Probability of Event P(E) = No. of. times that event occurs/ Total number of trials

The basic difference between these two approaches is that in the experimental approach; the probability of an event is based on what has actually happened by conducting a series of actual experiments, while in theoretical approach; we attempt to predict what will occur without actually performing the experiments.

Elementary Event

An event having only one outcome of the experiment is called an elementary event.

Example: Take the experiment of tossing a coin n number of times. One trial of this experiment has two possible outcomes: Heads(H) or Tails(T). So for an individual toss, it has only one outcome, i.e Heads or Tails.

Sum of Probabilities

The sum of the probabilities of all the elementary events of an experiment is one.

Example: take the coin-tossing experiment. P(Heads) + P(Tails)

= (1/2)+ (1/2) =1

Impossible event

An event that has no chance of occurring is called an Impossible event, i.e. P(E) = 0.

E.g: Probability of getting a 7 on a roll of a die is 0. As 7 can never be an outcome of this trial.

Sure event

An event that has a 100% probability of occurrence is called a sure event. The probability of occurrence of a sure event is one.

E.g: What is the probability that a number obtained after throwing a die is less than 7?

So, P(E) = P(Getting a number less than 7) = 6/6= 1

Range of Probability of an event

The range of probability of an event lies between 0 and 1 inclusive of 0 and 1, i.e. 0≤P(E)≤1.

Geometric Probability

Geometric probability is the calculation of the likelihood that one will hit a particular area of a figure. It is calculated by dividing the desired area by the total area. In the case of Geometrical probability, there are infinite outcomes.

Complementary Events

Complementary events are two outcomes of an event that are the only two possible outcomes. This is like flipping a coin and getting heads or tails.

, where E and

are complementary events. The event, representing ‘not E‘, is called the complement of the event E.

For any event A, there exists another event A‘ which shows the remaining elements of the sample space S. A’ denotes complementary event of A.

A’ = S – A.

Event A and A’ are mutually exclusive and exhaustive.

Consider the example of tossing a coin. Let P(E) denote the probability of getting a tail when a coin is tossed. Then probability of getting a head is denoted by

The eventmeans ‘Not E’.

Example 1:

A bag contains only lemon-flavoured candies. Arjun takes out one candy without looking into the bag. What is the probability that he takes out an orange-flavoured candy?

Solution:

Let us take the number of candies in the bag to be 100.

Number of orange flavoured candies = 0 [since the bag contains only lemon-flavoured candies]

Hence, the probability that he takes out an orange-flavoured candy is:

P (Taking orange-flavoured candy) = Number of orange flavoured candies / Total number of candies.

= 0/100 = 0

Hence, the probability that Arjun takes out an orange-flavoured candy is 0.

This proves that the probability of an impossible event is 0.

Example 2:

A game of chance consists of spinning an arrow that comes to rest pointing at any one of the numbers such as 1, 2, 3, 4, 5, 6, 7, 8 and these are equally likely outcomes. What is the probability that it will point at (i)8, (ii) Number greater than 2 (iii) Odd numbers.

Solution:

Sample Space = {1, 2, 3, 4, 5, 6, 7, 8}

Total Numbers = 8

(i) Probability that the arrow will point at 8:

Number of times we can get 8 = 1

P (Getting 8) = 1/8.

(ii) Probability that the arrow will point at the number greater than 2:

Number greater than 2 = 3, 4, 5, 6, 7, 8.

No. of numbers greater than 2 = 6

P (Getting numbers greater than 2) = 6/8 = 3/4.

(iii) Probability that the arrow will point at the odd numbers:

Odd number of outcomes = 1, 3, 5, 7

Number of odd numbers = 4.

P (Getting odd numbers) = 4/8 = ½.