# Class 10 – Probability

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 Introduction:If we go to North Pole or Mount Everest, we are sure to find snow there. Similarly, if we travel down to a seashore, what are we sure to find there? It is sand. On the other hand, there are some events about which we are not sure or certain like the following: Will rains be good this year? Will I get good marks in my next math’s test? Will I be able to get home on time to attend the party? Who will be the next world cup cricket champions? Also, if we toss a coin we cannot say for certain whether the head will show up or the tail will show up. The outcomes of such actions, as tossing a coin, throwing up a die, or drawing a card from a well shuffled pack of cards all depend on CHANCE. Chance: Plays an important part in life and many important decisions depend on the result of such action. Thus in a cricket match which team will have the right to decide whether to bat or field first is decided by the toss of a coin. Quality of goods purchased, for example, sweets, are determined by checking a sample, like tasting a single piece. Gamblers always depend on chance and make or ruin fortunes. In fact, the theory of probability was first applied to gambling. Some basic terms and concepts: a)              Experiment: A process which results in some well-defined outcomes is known as an experiment. Examples: 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. Examples: 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 of the experiment is not known in advance.
Examples:

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 to know 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.

Examples:

i)               When we toss a coin once, it may come up 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 experiment is getting any of the six number 1, 2, 3, 4, 5 and 6. Hence the sample space for 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 of them, out of which there are 26 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 tail up.

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

In case of throwing a dice:

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 each of the numbers 1, 2, 3, 4, 5 and 6 has the same possibility to come to the upper-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.

The probability scale: Mathematically, an event that will not happen has a probability zero. An event that is sure to occur will have a probability 1.

If a die is rolled, the probability of getting 1, 2, 3, 4, 5 or 6 is 1 as one of the numbers is certain to show up.

However, the probability of getting a 7 is zero as this is impossible.

This shows that all probabilities must have a value greater than or equal to 0 and less than or equal to 1. This can be shown on probability scale.

Probability is a way of measuring the chance of an event occuring.

We calculate the probability P (E) of the occurrence of an event E by the formula

where,

i)               Outcome is the result of an experiment.

ii)             Favorable outcome is an outcome that matches the event.

We can understand it better by the following examples:

 Activity Possible Outcomes Probability of Event 1. Flipping a coin Head, Tail 2. Throwing a die 1, 2, 3, 4, 5, 6 3. Spinning a spinner A, B

1. The probabilities of all possible outcomes of an event add up to 1.

2. P (event not happening) = 1 – P (event happening)

Example: A spinner having the number 2, 3, 4, 5, 6, and 7 is spun. What is the probability of getting:

(i) a 4?

(ii) a prime number ?

(iii) an even number ?

(iv) not a 5.

Solution:  Probability of an event

(i) Total number of possible outcomes = 6 (2, 3, 4, 5, 6, 7)

Chance of getting a 4 = 1

(ii) The prime numbers are 2, 3, 5, 7

Number of favourable outcomes = 4

Total number of possible outcomes = 6

.

(iii) The even numbers are 2, 4, 6,

Number of favorable outcomes = 3

Total number of possible outcomes = 6

(iv)

Example:Instead of numbers, the letters in the word CHANCE were stuck on a die. Find the probability of rolling:

(i) Letter H                  (ii) a vowel                   (iii) a consonant           (iv) any letter except E

Solution: A die has six faces and the letters C, H, A, N, C, E are stuck on one face each.

Total number of possible 6 (C, H, A, N, C, E)

Also, Probability of an event

i)              The letter H appears on one face

Number of favourable outcomes = 1

Total number of possible outcomes = 6

ii)             The vowels in the word CHANCE are A and E.

Number of favourable outcomes = 2

Total number of possible outcomes = 6

iii)           The consonants in the word CHANCE are C, H, N, C.

Number of favorable outcomes = 4

Total number of possible outcomes = 6

.

iv)           The letters except E are C, H, A, N, C.

Number of favourable outcomes = 5

Total number of possible outcomes = 6

.

Example: A card is drawn from a pack of 100 cards numbered 1 to 100. Find the probability of drawing a number which is a perfect square.

Solution: Out of the numbers 1 to 100, square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.

Thus there are 10 perfect square numbers.

Total number of possible events = 100

The number of favourable events = 10

Probability of drawing a card which is a square number.

Example: A standard pack of playing cards consists of 52 cards.

i)               How many red cards are there? Find the probability of drawing a red card from the well shuffled pack of cards?

ii)             How many hearts are there? Find the probability of drawing a heart?

iii)             What is the probability of drawing a queen?

iv)            What is the probability of drawing a black jack?

v)             What is the probability of drawing a queen or a black jack?

Solution: Total number of possible outcomes = 52 (as there are 52 cards)

(i) Number of red cards = 26

Number of favourable outcomes = 26

Total number of possible outcomes = 52

(ii) Number of hearts = 13

Number of favourable outcomes = 13

Total number of possible outcomes = 52

.

(iii) Number of favorable outcomes of a queen = 4

Total number of possible outcomes = 52

.

(iv) There are two black jacks (one club + one spade)

Number of favourable outcomes = 2

Number of possible outcomes = 52

(v) P (Queen or a black jack)

There are ‘4’ queens and ‘2’ black jacks.

Number of favourable outcomes = 6

Number of possible outcomes = 52

Estimating Probability by experiments:Experimental probability is found by collecting a large set of results by performing an experiment again and again.

Example: Ananya tossed a coin 10 times and found that out of the 10 times she got a head as 3 outcomes only. This shows that:

She repeated this experiment of 10 trials again and the probability of tossing a head

came out to be.

The probability thus calculated is the experimental probability. It is calculated by dividing the number of times the event occurs by the total number of trials.

We know that the probability of tossing a head when a coin is tossed  or in this case. This is called theoretical probability which is found by saying that all the events are equally likely.

Comparing the probability calculated by experiment and the theoretical probability, we see that there is a big difference between them.

The number of trials in the experiment were increased to 1000 times and Ananya found that she got 515 heads and 485 tails this time. This, as you can see, is much closer to the theoretical probability which in this case would be. If we repeat this many times, the average number of heads (or tails) would be close to 500.

Activity:

i) Toss a coin 40 times and record the results in a table.

 Outcomes Heads Tails Tally Marks Total

ii) Calculate the experimental probabilities for your experiment.

iii) What is the theoretical probability of getting a head on a single toss?

iv) How many of your 40 trials give you heads as a result?

v) Were your experimental probabilities close to the theoretical probabilities?

vi) Increase the number of trials to 60 and see what the change is.

The more the number of trials are in a probability experiment, the closer the experimental probability would agree with the theoretical probability.