Innings2
Create New Account

Sign in to Innings2

or

New Account     Reset Password     

Probability

    Introduction

    Probability

    Uses of Probability in Real Life

    Exercise 14.1

    What We Have Discussed

    Enhanced Curriculum Support

    Easy Level Worksheet

    Moderate Level Worksheet

    Hard Level Worksheet

Powered by Innings 2

Reset Progress

This will delete your progress and chat data for all chapters in this course, and cannot be undone!

Glossary

Select one of the keywords on the left…

Probability > Introduction

Introduction

Probabilities and likelihoods are everywhere around us, it is used in:

(1) Weather forecasting

(2) Games

(3) Insurance

(4) Election polls etc.

However, in the history of mathematics, probability is actually a very recent idea. While numbers and geometry were studied by ancient Greek mathematicians more than 2500 years ago, the concepts of probability only emerged in the 17th and 18th century.

According to legend, two of the greatest mathematicians, Blaise Pascal and Pierre de Fermat, would regularly meet up in a small cafe in Paris.

To distract from the difficult mathematical theories they were discussing, they often played a simple game: they repeatedly tossed a coin every heads was a point for Pascal and every tails was a point for Fermat. Whoever had fewer points after three coin tosses had to pay the bill.

One day, however, they get interrupted after the first coin toss and Fermat has to leave urgently.

Later, they wonder who should pay the bill, or if there is a fair way to split it.

The first coin landed heads (a point for Pascal), so maybe Fermat should pay everything. However, there is a small chance that Fermat could have still won if the had been tails.

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 on 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

Pascal and Fermat decided to write down all possible ways the game could have continued:

HHH

wins

HHT

wins

HTH

wins

HTT

wins

All four possible outcomes are equally likely, and Pascal wins in of them.

Thus, they decided that Fermat should pay 34 of the bill and Pascal should pay 14.

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 = .

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).

In general, it is true that for an event E:

P(E‾) = 1 – P(E)

The event E , representing ‘not E’, is called the complement of the event E. We also say that E and E‾ are complementary events.

Before proceeding further, let us try to find the answers to the following questions:

(i) What is the probability of getting a number 8 in a single throw of a die?

To get number 8 in a single throw, we need to have a number 8 on the die which isn't possible. That is, the probability of an event which is impossible to occur is .

Such an event is called an impossible event.

(ii) What is the probability of getting a number less than 7 in a single throw of a die?

Since every face of a die is marked with a number less than 7, it is sure that we will always get a number less than 7 when it is thrown once. So, the number of favourable outcomes is the same as the number of all possible outcomes, which is .

Therefore, P(E) = P(getting a number less than 7) = Number of favourable outcomesTotal number of outcomes= 66 =

For an event, where the probability was 1(getting a number less than 7) it is known as an sure event or certain event.

Note : From the definition of the probability P(E), we see that the numerator (number of outcomes favourable to the event E) is always than or equal to the denominator (the number of all possible outcomes). Therefore,

≤ P(E) ≤

Some additional Information

Predicting the Future

If we roll a die, the result is a number between 1 and 6, and all outcomes are equally likely. If we roll two dice at once and add up their scores, we can get results from up to . However, in this case they are not all equally likely.

Some results can only happen one way (to get 12 you have to roll + ) while others can happen in multiple different ways (to get 5 you have to roll either + or + ).

This table shows all possible outcomes:

2
3
4
5
6
7
8
9
10
11
12

The most likely result when rolling two dice is 7. There are outcomes where the sum is 7, and outcomes in total, so the probability of getting a 7 is 636 = .

The least likely outcomes are 2 and 12, each with a probability of 136 = .

It is impossible to forecast the outcome of a single coin toss or die roll. However, using probability we can very accurately predict the outcome of dice.

If we throw a die 30 times, we know that we would get around 16 × 30 = sixes.

If we roll it 300 times, there will be around 16 × 300 = sixes. These predictions get more and more accurate as we repeat the predictions more and more often.

In this animation you can roll many “virtual” dice at once and see how the results compare to the predicted probabilities:

Rolling Dice

We roll ${d} dice at once and record the SUM of their scores. The green lines represent the probabilities of every possible outcome predicted by probability theory and the blue bars show how often each outcome happened in this computer generated experiment.

Notice how, as we roll more and more dice, the observed frequencies become closer and closer to the frequencies we predicted using probability theory. This principle applies to all probability experiments and is called the law of large numbers.

Similarly, as we increase the number of dice rolled at once, you can also see that the probabilities change from a straight line (one die) to a triangle (two dice) and then to a “bell-shaped” curve. This is known as the central limit theorem, and the bell-shaped curve is called the normal distribution.