Probability

Probability is a way of expressing knowledge or belief that an event will occur or has occurred. The concept has an exact mathematical meaning in probability theory, which is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, artificial intelligence/machine learning and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.

SAMPLE SPACE

An experiment is a process or an operation with an outcomes.

The set of all possible outcomes of an experiment is called the sample space. It usually denoted by S.

When toss the coin, we can get only 2 results:

•     Head
•  Tail        

Example 1:

En. Adam has a fruit stall that sells bananas, apples,watermelons, papayas and durians. Students of class 4KP are asked to select their favorite fruit from the fruits at En. Adam’s stall.

S = { banana, apple, watermelon, papaya, durian}

Example 2:

A month is randomly selected from a year. Describe the sample space of this experiment by using set notation.

S= { January, February, March, April, May, June, July, August, September, October, November, December}

EVENT

•   Is a subset of the sample space.
•   Is an outcome or a set of outcomes that satisfies certain condition.
•   Denoted by a capital letter.

Example 1:

A box contains five cards written with 1,2,3,4 and 5 respectively. A card is picked randomly from the box.

S = {1, 2, 3, 4, 5}.

If we define J as ‘the card with  J as ‘the card with an even number’ ,the outcome of J in set notation will be J = { 2, 4 }.

J is known as an event of the experiment.

The number of outcome of an event n(P)=2

Example 2:

A letter is randomly selected from the word ‘COMPUTER’. Determine the number of possible outcomes of the event that the selected letter is

i. A vowel

ii. A consonant

Solution :

Let A = event that the selected letter is vowel = {O, U, E}

Therefore n (A) = 3

Let B = event that the selected letter is consonant = {C, M , P, T, R} Therefore n (B) = 5

PROBABILITY OF AN EVENT

P(E) = number of outcomes of the event

           _________________________________

           number of outcomes of the sample space

P(E) = n (E)

          = n (S)

P(E) = 0 means that it is impossible for the event to happen.

P(E) =1 means that the event is certain to happen.

The closer the probability of a given event is to 1, the more likely it is to happen.

Example

A bag contains 3 red balls and 4 white ones. If Rashid puts his hand in the bag and picks a ball, what is the probability that the ball he picked is white?

Solution:

S = {R1,R2,R3,W1,W2,W3,W4}

n(S)= 7

Let E is the event of drawing a white ball

E = {W1,W2,W3,W4}

n(E)=4

Therefore, the probability of drawing a white ball is 4/7

by Jannathul