Probability of an event

Probability is widely used in mathematics, science, economics, data analysis, and everyday decision-making to quantify uncertainty.

Probability is a measure of how likely an event is to happen.

It is expressed as a number between 0 and 1

• 0 means the event is impossible.

• 1 means the event is certain.

• Values between 0 and 1 indicate varying degrees of likelihood.

Probability is calculated as:

Probability  = (Number of favorable outcomes, n(E)) / (Total number of possible outcomes, n(S))

For example, when rolling a fair six-sided die:

• Total possible outcomes = 6 (1, 2, 3, 4, 5, 6)

• Favorable outcomes for rolling a 4 = 1

Prolling a 4  =  1/6

Sample Space and Events

• Sample Space (S) – the set of all possible outcomes of an experiment.

• Event (E) – a subset of the sample space containing outcomes of interest.

Tossing a Coin

Sample Space:

       S = {H, T}

Events:

Event A: Getting a Head

             A = {H}

Event B: Getting a Tail

                 B = {T}

Rolling a Die

Sample Space:

       S = {1, 2, 3, 4, 5, 6}

Events:

• Event A: Rolling an even number

                           A = {2, 4, 6}

• Event B: Rolling a number greater than 4

                           B = {5, 6}

• Event C: Rolling a prime number

                           C = {2, 3, 5}

Choosing a Day of the Week

Sample Space:

       S = {Mon, Tue, Wed, Thu, Fri, Sat, Sun}

Events:

• Event A: Choosing a weekend day

                           A = {Sat, Sun}

• Event B: Choosing a weekday

                           B = {Mon, Tue, Wed, Thu, Fri}

Calculating the Probability of an Event

1. By Listing Outcomes

Example 1: Rolling a Die

What is the probability of rolling an even number?

Step 1: List the sample space

       S = {1, 2, 3, 4, 5, 6}

Step 2: List the favorable outcomes

       Event E = rolling an even number

       E = {2, 4, 6}

Step 3: Count the outcomes

Favorable outcomes = 3

Total outcomes = 6

PE = 3/6

PE =  1/2

Example 2: Tossing Two Coins

Find the probability of getting exactly one head.

Step 1: List the sample space

       S = {HH, HT, TH, TT}

Step 2: List the favorable outcomes

       Event E = getting exactly one head

       E = {HT, TH}

Step 3: Count the outcomes

Favorable outcomes = 2

Total outcomes = 4

PE = 2/4

PE = 1/2

2. By Using Counting Techniques

Example 3: Choosing a committee

A committee of 2 students is chosen from 5 students.

What is the probability that a particular student, Ana, is included?

Step 1: Count all possible committees, Sample Space

n(S) = 5C2

n(S) = 10

Step 2: Count favorable committees, Events

n(E) = 4C1

n(E) = 4

Step 3: Calculate probability

PE = (n(E)) / (n(S)) 

PE = 4/10

PE = 2/5

Example 4: Drawing Cards

A card is drawn from a standard 52-card deck.

What is the probability of drawing an Ace?

Step 1: Count all Card, Sample Space

n(S) = 52

Step 2: Count Number of Aces, Events

n(E) = 4

Step 3: Calculate probability

PE = (n(E)) / (n(S)) 

PE = 4/10

PE = 2/5