Class 9 – Probability

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 Introduction: In everyday life, we come across statements such as i)      It will probably rain today. ii)     I doubt that he will pass the test. iii)   Most probably, Kavita will stand first in the annual examination. iv)   Chances are high that the prices of diesel will go up. v)     There is 50-50 chance of India winning a toss in today’s match. The words ‘probably’, ‘doubt’, ‘most probably’, ‘chances’, etc., used in the statements above involve an element of uncertainty. For example, in i), ‘probably rain’, will mean it may rain or may not rain today. We are predicting rain today based on our past experience when it rained under similar conditions. Similar predictions are also made in other cases listed in ii) to v). The uncertainty of ‘probably’ etc can be measured numerically by means of ‘probability’ in many cases. Though probability started with gambling, it has been used extensively in the fields of physical sciences, commerce, biological sciences, medical sciences, weather forecasting, etc. Probability: The branch of mathematics which deals with the estimation of chance is called the probability. The study of the theory of probabilities is of great mathematical interest and of great practical importance.   Some basic terms and concepts: a)    Experiment: A process which results in some well-defined outcomes is known as an experiment. For example: i)      When a coin is tossed, we shall be getting either a head or a tail i.e., its outcome is a head or a tail, which is well-defined. ii)     When a die is thrown the possible outcomes are 1, 2, 3, 4, 5 and 6, which are also well-defined. b)    Outcome: Result of an experiment is called an outcome. A random experiment may result in two or more outcomes. for example: i)      Tossing a coin. ii)     Throwing a dice, etc.

c) Random Experiment: Random experiment means all the outcomes of the experiment are known in advance, but any specific outcome ofthe experiment is not known in advance.
For example:

i) Tossing a coin is a random experiment because there are only two possible outcomes, head and tail, and these outcomes are known well in advance. But the specific outcome of the experiment i.e., whether a head or a tail is not known in advance.

ii) Throwing a die is a random experiment because we know in advance that there are only six possible outcomes of the experiment i.e., 1, 2, 3, 4, 5 and 6. But it is not possible toknow which of these six numbers will finally be the result.

Sample space: The set of all possible outcomes of an experiment is called sample space and is, in general, denoted by letter S.

For example:

i) When we toss a coin once, it may comeup in either of two ways: Head (H) or Tail (T). So, there are two possible outcomes of this random experiment. Thus the sample space (S) of this random experiment is given by S = {H, T}.

ii) When we roll a dice once, it may land with any of its 6 faces pointing upward. Thus, the outcome of this experimentis getting any of the six number 1, 2, 3, 4, 5 and 6. Hence the sample spacefor the experiment is S = {1, 2, 3, 4, 5, 6}.

iii) When two coins are tossed together, the random experiment may result:

a) Head (H) on the first coin and head (H) on the second coin.

b) Head (H) on the first coin and tail (T) on the second coin.

c) Tail (T) on the first coin and head (H) on the second coin.

d) Tail (T) on the first coin and tail (T) on the second coin.

Thus the corresponding sample space S ={(H, H), (H, T), (T, H), (T, T)}

iv) In a deck of cards there are 52 ofthem, out of which there are 2.6 red cards and 26 black cards. There are 3 picture cards in each set i.e., Jockey, Queen and king each set consists of 13 cards.

Equally likely outcomes:

In case of tossing a coin:

i) It is known, in advance that the coin will land with its head or tail up.

ii) It is reasonably assumed that each outcome, a head or a tail, is as likely to occur as the other. In other words,we say that there are equal changes for the coin to land with its head or tailup.

Referring to the terms used in this chapter, we say that the outcomes, head and tail, are equally likely.

In case of throwing adice:

i) It is known, in advance that the dice will show the number 1, 2, 3, 4, 5 or 6 on the upper-most face.

ii) It can reasonably be accepted that eachof the numbers 1, 2, 3, 4, 5 and 6 has the same possibility to come to theupper-most face.

Hence, showing up the numbers 1, 2, 3,4, 5 and 6 on the throwing of a die are equally likely outcomes.

Are the outcomes of all experiments equally likely?

Suppose a bag contains 6 red and 2 yellow balls. Let a ball be drawn from the bag without looking into it. The ball that will come out will either be a red or a yellow ball.

Are the outcomes, a red ball and yellow ball, equally likely?

No.

Reason: Since the bag contains 6 red and 2 yellow balls, then in a single draw of a ball from this bag (without looking into it);it is more likely to get a red ball than a yellow ball. Hence, the outcomes are not equally likely.
However, if the bag contains equal numbers of red balls and yellow balls, the outcomes are equally likely.

Event: A subset E of the sample space of all possible outcomes of an experiment is

called event.

Example: A die is rolled and the event be the“score is even”.

Solution: If a die is rolled then the sample space is S = {1, 2, 3,4, 5, 6}.
Let E be the event that score is even, then E = {2, 4, 6}

Definition ofprobability: If in a random experiment, the total number of events (outcomes) are n out of which m events (outcomes) are favourable to a particular event E then the probability of happening of event E is denoted by P(E) and is equal to the
ratio .

P(E) = Probability of the happening of event E =

Example: If a dice is rolled once, and an even number is required on the upper face of it, then in this experiment:

Total number of outcomes = 6 (any of 1,2, 3, 4, 5 and 6)
number of favourable outcomes = 3 (any of 2, 4 and 6)

Probability of getting an even number on the upper face.

a) Emperical (or)experimental probability:When the probability is based on an actual experiment, it is called an emperical (or) experimental probability.

Example: If a coin is tossed 100 times and the outcomes of this expriment are 57 heads and 43 tails, the probability of a headis and that of a tailis . Since these probabilities are based on the actual experiment of tossing a coin 100 times,they are experimental (or) emperical probabilities.

i) For finding the experiment probability,an adequate recording of the outcomes is required.

ii) Experimental probabilities are only‘estimates’. If the same experiment of tossing a coin 100 times is performed again, it will not necessarily give the same results of getting the number of heads and the number of tails. And so the probabilities for a head and a tail will also not be the same.

b) Classical (or)theoretical probability: When a repetition of an experiment can be avoided for calculating the exact probability, the probability so obtained is called classical (or) theoretical probability.

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

ii) In theoretical (classical) probability, we make certain assumptions and one of these assumptions is that the outcomes are equally likely.

iii) Probability of anevent (outcome)

iv) In this chapter,the probability means theoretical (or) classical probability.

Example: Find the probability of gettinga head when a coin is tossed once.

Solution: In the random experiment of tossing a coin once, the total number of possible outcomes is 2 which are Head (H) andTail (T).

Favourable outcome is ‘getting a head’

_Numberof favourable outcome = 1

P(getting a head)

An event, havingonly one favourable outcome, is called an elementary event.

Example: A bag contains a black ball, a red ball and a green ball, all the balls are identical in shape and size. Mohit takes out a ball from the bag, without looking into it. What is the probability that the ball drawn is:

(i) Red ball? (ii)Black ball? (iii)Green ball?

Solution: When Mohit takes out a ball without looking into the bag, the outcomes of the experiement are equally likely.

Clearly, the total number of possible outcomes = 3

(i) The number of favourable outcome (getting a red ball) = 1

_The probability of drawing a red ball.

In short:P(red ball) =

Similarly,(ii) P(drawing a black ball) =

And, (iii)P(getting a green ball) =

The sum of the probabilities of all the elementary events of an experiment is always one. In the example, given above.

P(red ball) +P(black ball) + P(green ball) = ++ = 1

Complementary events: For any event E, the event of non-occurence of E is called its complementary event and is denoted by (not E).

E and (not E) are called complementary events.

The sum of probabilities of an eventand its complementary event is always 1.

P(E) + P() = 1 _ P() = 1 – P(E)

Example: In a single throw of a die, find the probability of getting a number.

(i) Greater than 2 (ii)Less than or equal to 2 (iii)Not greater than 2.

Solution: In a single throw of a die, the totalpossible outcomes are 6(1, 2, 3, 4, 5 and 6)

i) Out of 1, 2, 3, 4, 5 and 6 ; thenumbers greater than 2 are 3, 4, 5 and 6
Total number of favourable outcomes = 4(3, 4, 5 and 6)
_P(getting a number greater than 2)

ii) Out of all possible outcomes 1, 2, 3,4, 5 and 6, the numbers less than or equal to 2 are 1 and 2
Total number of favourable outcomes = 2
_P(getting a number less than or equal to 2) =

iii) Out of all possible outcomes 1, 2, 3,4, 5 and 6, the numbers not greater than 2 is 1.
The number of favourable outcomes = 1
_P(getting a number not greater than 2) =

By observation:

i) In a single throw of a die, getting a number less than or equal to 2 and getting a number not greater than 2 means the same.

For this reason: P(getting a number less than or equal to 2)

=P(getting a number not greater than 2)

ii) If the event of getting number greater than 2 is denoted by E.

Then the event of getting a number not greater than 2 (or, a number less than or equal to 2) is denoted by not E or by
Thus P(E)= P(getting number greater than 2)
And, P() = P(not E) =P(getting number not greater than 2) =
P(E) + P() = + = 1
Also, P(E) + P() =1 _ P()= 1 – P(E)
i.e., P(not E) = 1 – P(E)

Impossible event: If the probability of an event is ‘0’ then the event is called an impossible event.

Example: In a single throw of a die, find the probability of getting 7.

Solution: When a die is thrown, the possible outcomes are 1, 2, 3,4, 5 and 6.

Since no face of the die has the number 7 marked on it, there is no outcome favourable to 7.

_The number of favourable outcomes = 0

_ P(getting a number 7)

Sure event: If the probability of an event is ‘1’then the event is called a certain event or a sure event.

Example: In a single throw of a die, find the probability of getting a number less than ‘7’.

Solution: When a die is thrown, the possible outcomesare 1, 2, 3, 4, 5 and 6

Every face of a die is marked with a number less than 7.

The number of favourable outcomes = the number of faces = 6

_P(getting a number less than 7)

Probability of any event can never be less than ‘0’ or more than ‘1’.

If E be anyevent, then 0 ≤ P (E) ≤ 1

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