Chance and Probability

Probability revolves around assigning a numerical value to this degree of uncertainty. Or in other words, probability refers to uncertainty about the occurrence of an event. The event may or may not occur. For example, we have some of the real-life statements which we hear in our daily lives: My uncle may visit us today; There is a good chance that it may rain tomorrow; The school may probably take us to a picnic in June; Schools may reopen in March. These daily life statements use the terms like ‘probable’, ’may’, ’has good chance’, ’likely’ etc., so it is clear that there is no surety of occurrence, it may occur, or it may not occur, so this is called chance. Or in other words, a chance is a possibility of something happening. And in mathematics, probability is known as chance. The concepts of probability are frequently used in Statistics, engineering, data science, and hypothesis testing.

Probability Formula

• This formula is used to calculate the probability of an event to occur.

P(E) = Probability (occurrence of an event) =

Note: The value of probability ranges from 0 to 1.

• This formula is used to calculate the probability of a non-occurrence of an event

 = 1 – P(E)

Here,  is a probability of a non-occurring event E. It is also known as the complement of an event E.

Terminology

Let us understand some fundamental concepts associated with probability. 

1. Experiment: An event in which some well-defined outcome is expected is called an experiment.  

2. Outcome: The result of an experiment is called outcome. For example, head/tail are possible outcomes while tossing a coin.

3. Sample Space: A set of all possible outcomes is called sample space. For example, while tossing a coin, sample space S is S = {H, T}, where H refers to head and T refers to Tail.

4. Random experiment: A random experiment is an experiment whose outcome may not be predicted in advance. It may be repeated under numerous conditions. Examples: 

1. Tossing a coin – Head and tails are the possible outcomes.

2. Rolling a die – There are six possible outcomes,1, 2, 3, 4, 5, 6.  

3. Number of goals scored in a soccer match.

5. Equally Likely Outcomes: The equally Likely Outcomes refers to a condition when each outcome of an experiment is equally likely. In other words, each outcome has the same chance of occurring. For example, while tossing a fair coin, there are equal chances to get a head or a tail.

6. Likely chances to probability: Let us consider the following cases to understand likely chances to probability.

Case 1: Tossing a coin

While tossing a coin, the sample space = {H, T}. There are two possible outcomes Head and tail, since both the outcomes are equally likely, we can conclude that the likelihood of getting head = 1/2. Similarly, the likelihood of getting tail = 1/2

Case 2: Rolling a die:

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

Total possible outcomes = 6

Since all outcomes have the equal chance, so the likelihood of occurrence of each outcome = 1/6

7. Outcomes as events: The occurrence of each outcome in an experiment forms an event. For example, in the experiment of tossing a coin, the occurrence of the head, as well as the occurrence of a tail, are considered as events.

8. Impossible Event: When the probability of an event is 0, then the event is known as an impossible event.

9. Sure Event: When the probability of an event is 1, then the event is known as a sure event.

Applications of chances & probability in Real Life

1. Weather prediction: Metrological department observes the trends from data over many years to predict the weather.

2. Exit polls: Exit polls are carried out to get an idea of the chances of winning candidates.  

3. Strategic planning: Businessmen study the trends of sales in a period of time to predict future sales. This helps them to plan strategically to provide them more profits.

Sample Problems

Question 1. What is the probability of getting an odd number while rolling a die?

Solution:

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

Total number of outcomes = 6

Odd numbers = {1, 3, 5}

So, the number of outcomes that make the desired event = 3

Probability = number of outcomes that make the desired event/ Total number of outcomes

= 3/6  

Question 2.  A spinning wheel has 7 green sectors, 5 red sectors, and 4 blue sectors. Find the probability of getting a red sector. Also, find the probability of getting a non-red sectors.

Solution:

Number of green sectors = 7

Number of red sectors = 5

Number of blue sectors = 4

Total number of sectors = 7 + 5 + 4 = 16

Probability of getting a red sector = No. of red sectors / Total number of sectors

= 5/16

Probability of getting a non-red sector = 1 – Probability of getting a red sector

= 1 – 5/16 

=11/16

Question 3.  Find the probability of getting a multiple of 3 when a die is rolled.

Solution:

While rolling a die possible outcomes = {1, 2, 3, 4, 5, 6}

Total number of possible outcomes = 6

Multiples of 3 = {3, 6}

So, number of desired outcomes = 2

Probability of getting a multiple of 3 = number of desired outcomes / Total number of possible outcomes 

= 2/6 

= 1/3

Question 4. Find the probability of getting a card of kings from a deck of 52 cards.

Solution:

Total number of cards = 52

Number of king cards = 4

Probability = number of desired outcomes / Total number of possible outcomes 

= 4/52 

= 1/13

Question 5. Find the probability of picking vowels in the word ‘CHAMPION’

Solution:

Total number of letters in the word ‘CHAMPION’ = 8

Vowels in the word CHAMPION = A, I, O

So, number of desired outcomes = 3

Probability = number of desired outcomes / Total number of possible outcomes

= 3/8

Question 6.  A bag is filled with balls. Some of these balls are red in color. The probability of picking a red ball is x/2. Find “x” if the probability of picking a non-red ball is 2/3.

Solution:

P(red ball) + P(non-red ball) = 1

Given that P(red ball) = x/2

P(non-red ball) = 2/3

= x/2 + 2/3 = 1

= x/2 = 1 – 2/3

x/2 = 1/3

x = 2/3

Question 7. What is the probability of the sun rising in ‘west’?

Solution: 

It is a universal fact that sun rises in East.

So, probability of sun rising in west = 0

It is an impossible event.

Question 8.  There are 24 students in a class. Out of these, 24 students, 16 are boys and the remaining are girls. Find the probability of selecting a girl randomly.

Solution: 

Total number of students = 24

Boys = 16

So, girls = 24 – 16 = 8

P(selecting a girl) = number of girls/total number of students

= 8/24 = 1/3

Question 9. There are 20 defective bulbs in a box of 500 electric bubs. Find the probability of randomly selecting a non-defective bulb.

Solution:

Total number of bulbs = 500

Defective bulbs = 20

So, number of non-defective bulbs = 500 – 20 = 480

P(selecting a non-defective bulb) = number of non-defective bulbs/Total number of bulbs

= 480/500

= 24/25

Question 10. A survey was conducted on 800 people. It was observed that 450 persons liked tea while the remaining 350 liked coffee. If a person is randomly selected, what is the probability that he likes coffee?

Solution:

Total number of persons = 800

Number of persons who like coffee = 350

So, probability of a randomly chosen person to like coffee = 350/800

= 7/16