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Chapter 13: Probability

    Introduction

    Probability - A theoretical approach

    Mutually Exclusive Events

    Deck of Cards and Probability

    Use of Probability

    More applications of Probability

    Summary

    Enhanced Curriculum Support

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Chapter 13: Probability > Probability - A theoretical approach

Probability - A theoretical approach

Pascal and Fermat had discovered the first important equation of probability: if an experiment has multiple possible outcomes which are all equally likely, then:

Probability of an event = Number of ways the event could happenTotal number of possible outcomes.

In our example, the probability of Pascal winning the game is 34 = , and the probability of Fermat winning the game is 14 = .

Probability for winning in a dice game: For another example of equally likely outcomes, suppose we throw a die once.

For us, a die will always mean a fair die. What are the possible outcomes?

This table shows all possible outcomes of throwing a die:

X
1
2
3
4
5
6
1

They are 1, 2, 3, 4, 5, 6. Each number has the same possibility of showing up. So the equally likely outcomes of throwing a die are 1, 2, 3, 4, 5 and 6. Let us try some:

Probability of getting 1 on the die =

Probability of (getting an even number on the die) = Number of even numbers on the dieTotal number of outcomes

=

Probability in drawing a specific colour ball: Suppose that a bag contains 4 red balls and 1 blue ball, and you draw a ball without looking into the bag.

What are the outcomes? Are the outcomes — a red ball and a blue ball equally likely?

Since there are 4 red balls and only one blue ball, you would agree that you are more likely to get a red ball than a blue ball.

So, the outcomes (a red ball or a blue ball) are not equally likely.

However, the outcome of drawing a ball of any colour from the bag is equally likely. So, all experiments do not necessarily have equally likely outcomes.

Note: However, in this chapter, from now on, we will assume that all the experiments have equally likely outcomes.

We have already defined the experimental or empirical probability P(E) of an event E as:

P(E) = Number of trials in which the event happenedTotal number of trials

The empirical interpretation of probability can be applied to every event associated with an experiment which can be repeated a large number of times.

The requirement of repeating an experiment has some limitations, as it may be very expensive or unfeasible in many situations. Of course, it worked well in coin tossing or die throwing experiments.

But how about repeating the experiment of launching a satellite in order to compute the empirical probability of its failure during launching, or the repetition of the phenomenon of an earthquake to compute the empirical probability of a multistoreyed building getting destroyed in an earthquake?

In experiments where we are prepared to make certain assumptions, the repetition of an experiment can be avoided, as the assumptions help in directly calculating the exact (theoretical) probability.

The assumption of equally likely outcomes (which is valid in many experiments, as in the two examples above, of a coin and of a die) is one such assumption that leads us to the following definition of probability of an event.

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

P(E) = Number of outcomes favourable to ENumber of all possible outcomes of the experiment

,where we assume that the outcomes of the experiment are equally likely.

Let us find the probability for some of the events associated with experiments where the equally likely assumption holds.

Probability theory had its origin in the 16th century when an Italian physician and mathematician J.Cardan wrote the first book on the subject, The Book on Games of Chance. Since its inception, the study of probability has attracted the attention of great mathematicians. James Bernoulli (1654 – 1705), A. de Moivre (1667 – 1754), and Pierre Simon Laplace are among those who made significant contributions to this field.

Laplace’s Theorie Analytique des Probabilités, 1812, is considered to be the greatest contribution by a single person to the theory of probability.

In recent years, probability has been used extensively in many areas such as biology, economics, genetics, physics, sociology etc.

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

Finding the probability

  • In the experiment of tossing a coin once, the number of possible outcomes is — Head (H) and Tail (T)
  • Let E be the event ‘getting a head’. Thus, P(E) = P (head) =
  • Similarly, if F is the event ‘getting a tail’, then P(F) = P(tail) =
  • We have found the answers

2. A bag contains a red ball, a blue ball and a yellow ball, all the balls being of the same size. Kritika takes out a ball from the bag without looking into it. What is the probability that she takes out the (i) yellow ball? (ii) red ball? (iii) blue ball?

Finding the probability

  • Kritika takes out a ball from the bag without looking into it. So, it is equally likely that she takes out any one of them.
  • Let Y be the event ‘the ball taken out is yellow’, B be the event ‘the ball taken out is blue’, and R be the event ‘the ball taken out is red’. Now, the number of possible outcomes = .
  • So, P(Y) = Number of yellow balls taken out by KritikaTotal number of outcomes =
  • P(R) = Number of red balls taken out by KritikaTotal number of outcomes =
  • P(B) = Number of blue balls taken out by KritikaTotal number of outcomes =
  • We have found the answers.

Remarks :

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

In Example 1, both the events E and F are elementary events i.e. : P(E) + P(F) = .

Similarly, in Example 2, all the three events, Y, B and R are elementary events i.e.: P(Y) + P(R) + P(B) = .

2. Observe that the sum of the probabilities of all the elementary events of an experiment is . This is true in general also.

3. Suppose we throw a die 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 ?

Finding the probability

  • Let E be the event ‘getting a number greater than 4’. The number of possible outcomes is
  • That is: 1, 2, 3, 4, 5 and 6, and the outcomes favourable to E are 5 and 6. Therefore, the number of outcomes favourable to E is .
  • So, P(E) = P(number greater than 4) = 26 =
  • Let F be the event ‘getting a number less than or equal to 4’. So, the number of outcomes favourable to F is .
  • Outcomes favourable to the event F are 1, 2, 3, 4. Therefore, P(F) = 46 =
  • We have found the answers

Remarks : From Example 1, we note that

P(E) + P(F) = 12 + 12 = (1)

where E is the event ‘getting a head’ and F is the event ‘getting a tail’.

From (i) and (ii) of Example 3, we also get

P(E) + P(F) = 13 + 23 = 33 = (2)

where E is the event ‘getting a number 4’ and F is the event ‘getting a number ≤ 4’.

Note that getting a number not greater than 4 is same as getting a number less than or equal to 4, and vice versa.

In (1) and (2) above, is F the same as ‘not E’?

We denote the event ‘not E’ by E‾.

So, P(E) + P(not E) =

i.e. P(E) + P( E‾) = 1, which gives us P(E‾) = 1 – P(E).