Probability

Section 1: Introduction to Probability

1.1 What is Probability?

In our daily lives, we frequently use words like "possibly," "likely," "probably," "certain," and "impossible." Consider these statements:

• "It will probably rain today."
• "There is a chance I will score good marks in the exam."
• "The government's new policy might increase employment opportunities."
• "Both candidates have an equal chance of winning the election."

These statements express uncertainty. When we say something is "likely" or "unlikely," we are making a judgment about the chance of that event happening. But can we measure this chance mathematically? Yes! This mathematical measurement of chance is called Probability.

Since ancient times, humans have been estimating the likelihood of various events in their daily lives. From predicting weather to making business decisions, probability helps us make informed choices in the face of uncertainty.

Definition: Probability is a branch of mathematics that deals with measuring the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain.

1.2 Key Definitions

To understand probability properly, we need to learn some fundamental terminology:

(a) Experiment

An experiment is any action or process that leads to well-defined outcomes. In probability, we perform experiments to observe the results.

Example:

• Tossing a coin
• Rolling a die
• Drawing a card from a deck
• Picking a ball from a bag

(b) Random Experiment 

A random experiment is an experiment where all possible outcomes are known, but the exact outcome of any single trial cannot be predicted with certainty.

Example:

• When you toss a coin, you know you'll get either Head (H) or Tail (T), but you cannot predict which one will appear in a particular toss.
• When you roll a die, you know you'll get one of {1, 2, 3, 4, 5, 6}, but you cannot predict the exact number.

(c) Outcome

An outcome is a possible result of a random experiment. Each trial of an experiment produces exactly one outcome.

Example:

• When tossing a coin, the outcomes are Head (H) or Tail (T).
• When rolling a die, the outcomes are 1, 2, 3, 4, 5, or 6.

(d) Sample Space 

The sample space is the set of all possible outcomes of a random experiment. It is usually denoted by the letter S.

Example:

• For a coin toss: S = {H, T}, and n(S) = 2
• For rolling a single die: S = {1, 2, 3, 4, 5, 6}, and n(S) = 6

(e) Event

An event is a subset of the sample space. It represents a specific outcome or a set of outcomes that we are interested in. Events are usually denoted by capital letters like E, A, B, etc.

Example:

• Getting a Head when tossing a coin: E = {H}
• Getting an even number when rolling a die: E = {2, 4, 6}
• Getting a King when drawing a card from a deck: E = {King of Hearts, King of Diamonds, King of Clubs, King of Spades}

(f) Equally Likely Outcomes

Outcomes are said to be equally likely if they have the same chance of occurring.

Example:

• When tossing a fair coin, the chance of getting Head and getting Tail are equal. So, H and T are equally likely outcomes.
• When rolling a fair die, each number from 1 to 6 has an equal chance of appearing.

Important Note: Not all outcomes are equally likely. For example, if a spinner has unequal sections, the outcomes are not equally likely.

(g) Mutually Exclusive Events

Two or more events are mutually exclusive if they cannot occur at the same time.

Example:

• In a single coin toss, getting Head and getting Tail are mutually exclusive events because you cannot get both in one toss.
• In a single die roll, getting an even number and getting an odd number are mutually exclusive.

(h) Favorable Outcomes

Favorable outcomes are the outcomes in the sample space that correspond to a particular event.

Example:

• Event E = getting an even number when rolling a die. Favorable outcomes = {2, 4, 6}.
• Event A = getting a red ball from a bag of colored balls. Favorable outcomes = the number of red balls.

(i) Elementary Event

An elementary event is an event that consists of exactly one outcome.

Example:

• Getting a 5 when rolling a die: E = {5}
• Getting a Queen of Spades when drawing a card: E = {Queen of Spades}

Section 2: Theoretical (Classical) Probability

2.1 Formula for Theoretical Probability

Theoretical probability is based on logical reasoning. If all outcomes in a sample space are equally likely, the theoretical probability of an event E is given by:

\(P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{n(E)}{n(S)} \)

Where:

• P(E) = Probability of event E
• n(E) = Number of outcomes favorable to event E
• n(S) = Total number of outcomes in the sample space

Rules of Probability:

Range of Probability: Probability of any event E is always between 0 and 1 inclusive. $ 0 \leq P(E) \leq 1 $

Impossible Event: If an event cannot occur, its probability is 0. $ P(\text{Impossible Event}) = 0 $

Certain Event: If an event is certain to occur, its probability is 1. $ P(\text{Certain Event}) = 1 $

Complementary Events: The probability of an event not occurring is 1 minus the probability of it occurring. $ P(\text{not } E) = 1 - P(E) $

2.2 Worked Examples

Example 1: Tossing a Coin

Problem: A fair coin is tossed once. What is the probability of getting:

• (i) A Head (H)
• (ii) A Tail (T)

Solution:

Sample space: S = {H, T}

• n(S) = 2

(i) Event E = Getting a Head = {H}

• n(E) = 1
• P(E) = n(E)/n(S) = 1/2

(ii) Event F = Getting a Tail = {T}

• n(F) = 1
• P(F) = n(F)/n(S) = 1/2

Answer: P(H) = 1/2 and P(T) = 1/2

Section 3: The Probability Scale

3.1 Understanding the Scale

The probability scale is a number line from 0 to 1 that helps us visualize and interpret probability values. It provides a visual way to understand whether an event is impossible, unlikely, equally likely, likely, or certain.

    0          1/4          1/2          3/4          1
    |-----------|------------|------------|-----------|
    |           |            |            |           |
Impossible   Unlikely    Equally     Likely     Certain
                          Likely

Interpreting Probabilities:

3.2 Worked Examples

Example 1: Spinner with Equal Sections

Problem: A spinner has 3 equal sections: Blue, Yellow, and Red. It is spun once. Find the probability of:

• (i) Landing on Blue
• (ii) NOT landing on Blue

Solution:

Total sections: n(S) = 3

(i) Event E = Landing on Blue

• n(E) = 1
• P(E) = 1/3

(ii) P(Not Blue) = 1 - P(Blue) = 1 - 1/3 = 2/3

Answer: P(Blue) = 1/3, P(Not Blue) = 2/3

Section 4: Empirical (Experimental) Probability

4.1 Introduction

While theoretical probability is based on logical reasoning, empirical probability (also called experimental probability) is based on actual experiments or observations. It is calculated by performing an experiment repeatedly and recording the results.

Formula:

\(\text{Empirical Probability of Event E} = \frac{\text{Number of trials where event E occurred}}{\text{Total number of trials}} \)

4.2 Comparison with Theoretical Probability

• Theoretical probability tells us what should happen in theory.
• Empirical probability tells us what actually happened in an experiment.
• The more trials we conduct, the closer the empirical probability usually gets to the theoretical probability. This is known as the Law of Large Numbers.

Example: If you toss a coin, theoretical probability says P(H) = 1/2 = 0.5.

• If you toss it 10 times, you might get 6 Heads (empirical P(H) = 6/10 = 0.6).
• If you toss it 100 times, you might get 52 Heads (empirical P(H) = 52/100 = 0.52).
• If you toss it 1,000 times, you might get 503 Heads (empirical P(H) = 503/1000 = 0.503).
• As the number of trials increases, the empirical probability gets closer to 0.5.

4.3 Worked Examples

Example 1: Coin Toss Experiment

Problem: A coin is tossed 50 times. The results are:

• Head (H) appeared 23 times
• Tail (T) appeared 27 times

Find the empirical probability of getting:

• (i) Head
• (ii) Tail

Solution:

Total trials = 50

(i) P(H) = Number of Heads / Total trials = 23/50

(ii) P(T) = Number of Tails / Total trials = 27/50

Answer: P(H) = 23/50, P(T) = 27/50

Chapter Summary

Key Takeaways

Definition: Probability is the mathematical measure of the likelihood of an event occurring.

Formula (Theoretical): [ P(E) = \frac{n(E)}{n(S)} ] This applies only when all outcomes are equally likely.

Formula (Empirical): [ P(E) = \frac{\text{Number of times E occurred}}{\text{Total number of trials}} ]

Probability Scale: Probability values range from 0 to 1:

• 0 = Impossible
• 1 = Certain
• 0.5 = Equally Likely

Complementary Events: $ P(\text{not } E) = 1 - P(E) $

Key Terms to Remember:

• Experiment
• Random Experiment
• Outcome
• Sample Space (S)
• Event (E)
• Equally Likely Outcomes
• Mutually Exclusive Events
• Favorable Outcomes
• Elementary Event

Important Concepts:

• Sum of probabilities of all elementary events in a sample space = 1
• Theoretical probability is based on reasoning, empirical probability is based on experimentation
• More trials lead to empirical probability approaching theoretical probability

Quick Revision Questions

Question 1: What is the sample space for tossing a coin?

• A) {H}
• B) {T}
• C) {H, T}
• D) {1, 2}
• Answer: C

Question 2: A fair die is rolled once. What is the probability of getting a number greater than 2?

• A) 1/6
• B) 2/6
• C) 3/6
• D) 4/6
• Answer: D (Favorable outcomes: 3, 4, 5, 6 → n(E) = 4, P(E) = 4/6 = 2/3)

Question 3: If P(E) = 0.3, what is P(not E)?

• A) 0.3
• B) 0.7
• C) 1.0
• D) 0.03
• Answer: B (P(not E) = 1 - 0.3 = 0.7)

Question 4: In a bag with 2 red, 3 blue, and 5 green balls, what is the probability of drawing a green ball?

• A) 1/2
• B) 1/5
• C) 2/10
• D) 3/10
• Answer: A (n(S) = 10, n(Green) = 5, P(Green) = 5/10 = 1/2)

Question 5: Which of the following is NOT a possible probability value?

• A) 0.5
• B) 1.2
• C) 0
• D) 3/4
• Answer: B (Probability must be between 0 and 1 inclusive, so 1.2 is not possible)

Question 6: How many Kings are there in a standard deck of 52 cards?

• A) 1
• B) 4
• C) 13
• D) 26
• Answer: B

Question 7: The probability of an impossible event is:

• A) 0
• B) 1
• C) 1/2
• D) Cannot be determined
• Answer: A

Question 8: When three coins are tossed, how many outcomes are in the sample space?

• A) 3
• B) 6
• C) 8
• D) 12
• Answer: C (2 × 2 × 2 = 8 outcomes)