## Probability

#### Concepts of Probability

Probability is a concept which measures the degree of uncertainty of the occurrence of an event numerically. In
experiments, there is always an uncertainty about whether a particular event would occur or not. Thus, probability
measures this possibility of occurrence and is assigned by numbers between 0 and 1.

If the occurrence of an event is certain, then the probability is 100% or 1, and if it is certain that an event will not
occur, then its probability is 0% or 0. The probability is between 0 and 1 if it is not sure whether the event will occur
or not.

Certain concepts used in Probability

1.     Experiment: An operation that can produce a well defined outcome is called an experiment. Example: Throwing
of a die.

2.     Sample Space: This is the set of all the possible outcomes of a random experiment. For example: sample space
of tossing a coin once is {Heads, Tails}.

3.     Event: An event is a subset of the sample space.

#### Multiplication and Addition Rule of Probability

When an event occurs one after the other in a single task then the probability of the task is the product of the
probabilities of each event that are part of the same task. This is called multiplication rule of probability. When the
events belong to a different task then their probabilities get added. This is called addition rule of probability.

Two or more events may occur independently or may depend on other events. For example, if the experiment is
about rolling dice and the event is getting an even number on the dice. Each event is independent of the other and
does not depend on the previous outcome. In this case, the probability of getting even number will be same for each
event.

Multiplication rule example

Example

A die is rolled thrice. Find the probability of getting an even number each time.

Solution

Sample space for experiment =all possible numbers of die = {1, 2, 3, 4, 5, 6}

Sample space for even number = {2, 4, 6}

Probability of getting even number on a die = 3/6 =1/2 or 0.5

Here the task is to get an even number thrice by rolling of the die.

Probability of getting an even number on each die = 0.5 x 0.5 x 0.5 = (0.5)3

Example

Two dice are rolled. What is the probability that the sum of the numbers on two dice is less than 4?

Solution

Sample space for experiment= all possible sums

 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12

Event = Sum is less than 4

It can be split into events that sum =2 or sum =3

Number of ways to get sum as 2 is 1 i.e. when both dice get 1

Number of ways to get sum as 3 is 2 which is (1+2) or (2+ 1)

Total number of possible combinations = 6 x 6 = 36

Probability of getting 2 =1/36

Probability of getting 3 = 2/36

Probability of event to get either 2 or 3 = Probability of getting 2 + Probability of getting 3

= 1/36 + 2/36 = 3/36

#### Applications of Probability

The main factors that determine the probability of developing lung cancer include duration of smoking, average
number of cigarettes smoked per day, duration of smoking abstinence, and age.

For example, a 50-year-old man who smoked a pack of cigarettes per day for 30 years but had not smoked for the
past ten years had a less than 1% chance of developing lung cancer in the next 10 years. A 70-year-old man who
smoked two packs of cigarettes per day and continues to smoke had a 15% chance of developing lung cancer within
10 years.

Mathematics has been used to understand and predict the spread of diseases.

Example

The probability that a smoker has lung cancer is very high, about .52, and that for a non smoker to have lung cancer
is only 0.08. If it is given that about 25% of the population smokes, what is the probability that a person having lung cancer
is a non smoker?

Probability that a person is a smoker = 0.25

Probability that a person is not a smoker = 0.75

Probability that a person is a smoker and has lung cancer = 0.52 x 0.25 = 13%

Probability that a person is a smoker and doesn't have lung cancer = 0.48 x 0.25 = 12%

Probability that a person is not a smoker and has lung cancer = 0.08 x 0.75 = 6%

Probability that a person is not a smoker and doesn't have lung cancer = 0.92 x 0.75 = 69%

Probability that a person has lung cancer = 13% + 6% = 19%

Given that a person has lung cancer, the probability that he is a non smoker = 6/19

#### Try these questions

1.     If a die is rolled once, what is the probability that it will show an odd number?

A.    0.5

B.    0.6

C.    0.7

D.    0.9

Solution

Total number of faces on a die = 6 numbered from 1 to 6

Number of faces with odd numbers = 3 for 1, 3 and 5.

2.     What are the possible outcomes if a coin is tossed once?

A.    (H,H)

B.    (H,T)

C.    (T,T)

D.    None of the above

*H is for head and T is for tail

Solution

When a coin is tossed, only two outcomes are possible i.e. either head (H) or tail (T).

3.    A bag has 2 blue balls and 7 yellow balls. Two balls are drawn at random. Find the probability function
of the number of blue balls drawn?

A.    0,1,2

B.    2,3,4

C.    1,2,3

D.    2,4,5

Solution

Step 1: Let x be the number of blue balls drawn.

Step 2: x = 0, 1 or 2

The number of blue balls drawn may be zero, one or both the balls.

4.     Probability that the next person you meet has a phone number that ends with 4?

A.    10%

B.    20%

C.    30%

D.    50%