The word 'Probability' is commonly used in our day-to-day conversation and we use this word even without going into the details of its actual meaning. Generally, people have a rough idea about its meaning. In our day-to-day life we come across statements like

      i) She is probably right

      ii) Probably it may rain today

      iii) He may possibly join politics

   In such statements, we generally use the terms probable, possible, chance, likely etc. All the terms convey the same sense that the event is not certain to take place or in other words, there is uncertainty about the happening of the event in question. In the theory of probability we assign numerical value to the degree of uncertainly.

What is probability?

Consider an experiment: A normal coin was tossed 1000 times. Head turned up 455 times and tail turned up 545 times. If we try to find the likelihood of getting heads we may say it is 455 out of 1000 or  or 0.455. 

This estimation of probability is based on the results of an actual experiment of tossing a coin 1000 times. These estimates are called experimental or empirical probabilities. In fact, all experimental probabilities are based on the results of actual experiments and an adequate recording of what happens in each of the events. The probabilities are only 'estimation'. If we perform the same experiments for another 1000 times, we may get slightly different data, giving different probability estimate. 

   Many other persons from different parts of the world have done this kind of experiment and recorded the number of heads that turned up. By these experiments we can generalize the experimental probability of a head (or a tail) may settle down closer and closer to the number 0.5 or . This matches we will learn how to find theoretical probability.

Theoretical Probability

Let us consider the following situation: Suppose a 'fair' coin is tossed at random. 

   When we speak of a coin, we assume it to be 'fair' that is, it is symmetrical so that there is no reason for it to come down more often one side than the other. We call this property of the coin as being 'unbiased'. By the phrase 'random toss', we mean that the coin is allowed to fall freely without any bias or interference. We refer to this by saying that the out comes, head and tail are equally likely.         

   For basic understanding of probability, we will assume that all the experiments have equally likely outcomes.

   Now, we know that the experimental or empirical probability P(E) of an event

                      

   The theoretical probability or classical probability of an event T, written as P(T) is defined as

                  

   Where we assume that the out comes of the experiments are equally likely. We usually simply refer to theoretical probability as probability.
 

Mind Map

Important Concepts

‣ Probability means an estimation.

‣ The definition of probability was given by Pierre Simon Laplace in 1795.

‣ Probability theory had its origin in 16th century.

‣ Italian physician and mathematician J. Cardan wrote the first book named as 'The Book on Games of Chance".

‣ James Bernoulli (1654 − 1705), A. De Moivre (1667 − 1754) and Pierre Simon Laplace (1749 − 1827) made a significant contribution to the theory of probability.

‣ The Probability estimated on the basis of results of an actual experiment is called experimental probability or empirical probability.

‣ Classical or Theoretical probability of an event (E) written P(E), is defined as

Definitions

Event: Event is an outcome satisfying given condition.

Sample space: Set of all outcomes is called sample space.

Equally likely events: Two or more events are said to be equally likely if each one of them has an equal chance of occurrence.

Mutually exclusive events: Two or more events are mutually exclusive if the occurrence of each event prevents the every other event.

Complementary events: Let 'E' be an event then the set 'F' of all outcomes which are against to 'E' is called its complementary event.

Exhaustive events: All the events are exhaustive events if their union is the sample space.

Sure event: The sample space of a random experiment is called sure or certain event as any one of its elements will surely occur in any trail of the experiment.

Impossible event: An event which will not occur on any account is called an impossible event.

Important Points

‣ The probability of an event 'E' is a number P(E) such that 0 ≤ P(E) ≤ 1.

‣ An event having only one outcome is called an elementary event. The sum of the probabilities of all elementary events of an experiment is '1' (one).

‣ The probability of a sure event is 1.

‣ The probability of an impossible event is zero.

Examples

1. Find the probability of getting a head when a coin is tossed once. Also find the probability of getting a tail.

Sol: If we toss a coin one time, the possible outcomes are two Head (H) and Tail (T)

∴ Sample space = {H, T}

Total number of out comes n(S) = 2

The number of out comes favourable to 'H' is n(H) = 1

Similarly, the number of out comes favourable to T is n(T) = 1

If we toss a coin two times then possible out comes are four.

Sample space = {(H, H), (H, T), (T, H), (T, T)}

Total number of out comes n(S) = 4

The number of out comes favourable to get at least one 'H' is n(H) = 3

Similarly,

Number of out comes favourable to get head only is 1.

Now we can generalize the situation

2. Suppose we throw a dice once.

i) What is the probability of getting a number greater than 4?

ii) What is the probability of getting a number less than or equal to 4?

Sol: i) In rolling an unbiased dice

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

No.of outcomes n(S) = 6

Favourable outcomes for number greater than 4 E = {5, 6}

No.of favourable outcomes n(E) = 2

ii) Let 'F' be the event getting a number less than or equal to 4.

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

No.of outcomes n(S) = 6

Favourable outcomes F = {1, 2, 3, 4}
No.of favourable outcomes n(F) = 4

 

3. Two dice are thrown at the same time. Write down all the possible outcomes. What is the probability that the sum of the two numbers appearing on the top of the dice is (i) 8 (ii) 13 (iii) less than or equal to 12?

Sol: When first dice shows 1, the second dice could show any one of the numbers 1, 2, 3, 4, 5, 6. The same is true when first dice shows 2, 3, 4, 5, 6.

The possible out comes are shown in the figure. The first number in each ordered pair is the number appearing on the first dice and second number is that on the second dice.

And note that (1, 4) is different from (4, 1)

so the number of possible out comes n(S) = 6 × 6 = 36

i) Let E be the event that the sum of the two numbers appearing on the top of the dice is 8.

Sample space for 'E' are {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}

∴ Number of outcomes favourable to E is n(E) = 5

ii) Let F be the event that the sum of two numbers appearing on the top of the dice is 13.

there is no out come for F

∴ Number of outcomes favourable to F is 0.

(Note that this event is called impossible event)

iii) Let G be the event that the sum of two numbers appearing on the top of the dice is less than or equal to 12.

As all the outcomes are favourable to the event 'G'.

∴ Number or outcomes favourable to G is 36.

(Note that this event is called sure event)

Impossible event: The probability of an event which is impossible to occur is '0'. Such an event is called an impossible event.

e.g.: Probability of getting 7 in a single throw of a dice.

Sure event: The probability of an event is sure to occur is 1. Such an event is called a sure event.

e.g.: Probability of getting 6 or a number less than 6 of a dice

Note: By the definition of probability P(E), we see that the numerator is always less than or equal to the denominator.

∴ 0 ≤ P(E) ≤ 1

Elementary event: An event has only one outcome is called an Elementary event

e.g.: If we toss a coin then getting a head or tail is an elementary event. Because it has only one outcome.

Counter: Probability of getting a number greater than 4 when we throw a dice is not an elementary event. Because this event has two outcomes (5 or 6).

Note that sum of probabilities of all elementary events is 1.
 

Examples

ii) If we throw a dice then

Complementary events:

   Let P(E) be the probability of an event. F is the all outcomes which are not favourable to E. We denote the event 'not E' by . This is called the complementary event of E.

4. One card is drawn from a well - shuffled deck of 52 cards. Calculate the probability that the card will

i) be an ace ii) not be an ace

Sol: Well shuffling ensures equally likely outcomes.

i) There are 4 aces in a deck let 'E' be the event ' the card is an ace' the number of favourable out comes to E = n(E) = 4

total number of outcomes = n(S) = 52

ii) Let 'F' be the event 'card not drawn is not an ace'.

The number of favourable outcomes to F = n(F) = 52 − 4 = 48

Total number of outcomes = n(S) = 52

Note that F is nothing but 

∴ We can also calculate P(F) = P() = 1 − P(E)
                                                              

∴ Here E and F are complementary events.
 

Uses of probability in daily life situations

1) We use probability in sports − the probability of winning a player or team.

2) We use probability in keep track of birthdays − same birth days etc.

3) We use probability in calculating how many (53) sundays (or tuesday or any day) appear in an ordinary year and in a leap year.
 

 

Writer: T.S.V.S. Suryanarayana Murthy