Introduction to Probability: The Mathematics of Maybe

Class 09 Maths

Probability is a type of measurement, similar to how we measure quantities like length, area, or volume. However, instead of measuring physical quantities, probability is used to measure the likelihood of events.

Probability is a measurement of the likelihood of an event.

There is an element of chance or randomness involved every time an event takes place.

What is Randomness?

Randomness refers to a situation or action (like the tossing of a coin or the rolling of a die) where you cannot predict exactly what will happen. Although you may know all the possible outcomes, you cannot say which one will definitely occur. For example,

  • Tossing a coin: You know it could be heads or tails, but you cannot be sure which one will come up in a single toss.
  • Rolling a die: You know the possible outcomes are 1, 2, 3, 4, 5, or 6, but you do not know which number will appear on a particular roll.

These observations (commonly called experiments or trials in probability) are called random because of their unpredictability.

Random Observations: A random experiment is something you can repeat (like tossing a coin), where every time you do it, the result might be different and you cannot know the outcome in advance.

Probability is the area of mathematics that studies randomness and how likely a specific outcome is to happen in a random situation. For example, when you toss a coin, the probability for heads is 1/2, and for tails is 1/2, because each is equally likely.

The Probability Scale

Probability is measured on a scale from 0 to 1 to indicate the likelihood of the occurrence of an event.

Probability is expressed on a scale from 0 (indicating impossibility) to 1 (indicating certainty), representing the likelihood of an event occurring. The probability of an event E is denoted as P(E), where 0 ≤ P(E) ≤ 1.

Measuring Probability Objectively

There are two main ways of estimating the probability of an event objectively.

Using evidence from experience: This involves collecting data either by performing an experiment multiple times or by analysing statistical data from past observations. In both cases, the probability is estimated by calculating the relative frequency of the event based on the collected evidence. We call this experimental probability.

Using theoretical methods: This approach assumes that all possible outcomes are equally likely. The probability is then determined by reasoning about the number of favourable outcomes in relation to the total number of possible outcomes. We call this theoretical probability.

Theoretical Probability

Theoretical probability studies the likelihood of an event happening based on all possible outcomes being equally likely. Theoretical probability is usually denoted as P (Event) or P (Outcome).

$$ \text{Theoretical Probability } (P) = \frac{\text{Number of favourable outcomes}}{\text{Number of possible outcomes}} $$

Example: If you roll a standard 6-sided die, what is the theoretical probability of getting a 4?

Number of favourable outcomes = 1 (only the number 4)

Number of all possible outcomes = 6 (the numbers 1 through 6)

P (rolling a 4) = 1/6 = 0.1666... ≈ 0.167 or 16.7%

Example: A letter is picked at random from the word ‘PROBABILITY’. What is the probability of picking the letter B?

Number of favourable outcomes = 2 (there are 2 Bs)

Number of all possible outcomes = 11 (The total number of letters)

P (Picking the letter B) = 2/11 = 0.1818... ≈ 0.182 or 18.2%.

Sample Space

The sample space, denoted by S, is the list of all possible outcomes. Each possible outcome is called an element of the sample space.

  • The sample space S must include every possible outcome.
  • No outcome should be listed more than once.
  • The number of elements in the sample space is called the sample size and is denoted by n(S).

Tossing a Coin

  • Experiment: Tossing a fair coin once.
  • Sample Space: S = {Heads (H), Tails (T)}
  • Sample Size = 2

Rolling a Die

  • Experiment: Rolling a standard 6-sided die.
  • Sample Space: S = {1, 2, 3, 4, 5, 6}
  • Sample Size = 6

Tossing Two Coins

  • Experiment: Tossing two coins simultaneously.
  • Sample Space: S = {HH, HT, TH, TT}
  • Sample Size = 4

Events

An event is any single possible result or combination of results that might happen when you perform a random action. It is like choosing particular outcomes from all the things that could possibly occur. An event is a subset of a sample space.

Tossing Two Coins

  • Sample Space = {HH, HT, TH, TT}
  • Event: At least one coin shows Head. E = {HH, HT, TH}

Rolling a 6-sided Die

  • Sample space: S = {1, 2, 3, 4, 5, 6}
  • Event: The number rolled is greater than 4. E = {5, 6}

Picking Fruit from a Basket

  • Sample space: Types of fruit you might pick, for example, Apple, Banana and Orange. S = {Apple, Banana, Orange}
  • Event: Picking a fruit that is yellow. E = {Banana}

Tree Diagrams

A tree diagram is a visual representation used to list all possible outcomes of a multi-step experiment. A multi-step experiment involves a series of independent trials. For example, tossing a coin two times, or rolling a dice three times are examples of multi-step experiments. Each branch of the tree represents a possible outcome, and branches split to show different paths for subsequent events.