Theoretical Probability — The Basic Idea

What Does Probability Measure?

Probability is a number that measures how likely an event is to happen. It runs from 0 (the event cannot happen) to 1 (the event is certain). A probability of $\dfrac12$ means an even chance — as likely to happen as not.

In Class 9 you estimated probability by experiment (tossing a coin many times and counting). Here we compute the exact, theoretical probability by reasoning about the possible outcomes — no experiment needed.

Equally Likely Outcomes and the Formula

The key assumption is that the possible outcomes are equally likely — none is favoured over another. When you toss a fair coin, head and tail are equally likely; when you throw a fair die, the six faces $1,2,3,4,5,6$ are equally likely.

For an event $E$, the theoretical probability is $$\boxed{P(E) = \dfrac{\text{number of outcomes favourable to } E}{\text{total number of equally likely outcomes}}}$$

How to use it: (1) list or count all equally likely outcomes (the sample space); (2) count how many are favourable to $E$; (3) divide.

Key Point: Every probability is (favourable)/(total). The whole chapter is careful counting.

Elementary Events

An elementary event is an event with exactly one outcome — for example, "getting a head" when tossing one coin, or "getting a 3" when throwing a die.

A neat fact ties them together:

The sum of the probabilities of all the elementary events of an experiment is 1.

For a die, $P(1)+P(2)+\dots+P(6) = \dfrac16\times 6 = 1$. For a coin, $P(H)+P(T)=\dfrac12+\dfrac12=1$.

[Board Important] Because all outcomes together are certain, their probabilities must total 1 — a quick way to check your counting.

Solved Examples

Example 1: One coin

A fair coin is tossed once. Find the probability of getting a head.

Solution:

1. Total equally likely outcomes: head, tail $\Rightarrow 2$.
2. Favourable to "head": $1$.
3. $P(\text{head}) = \dfrac12$.

Final Answer: $\dfrac12$.

Takeaway: Favourable over total.

Example 2: One die

A fair die is thrown once. Find the probability of getting (i) a 4, (ii) an even number.

Solution:

1. Total outcomes: $1,2,3,4,5,6 \Rightarrow 6$.
2. (i) Favourable to 4: just $\{4\}$, so $P = \dfrac16$.
3. (ii) Even numbers $\{2,4,6\}$: $3$ favourable, so $P = \dfrac36 = \dfrac12$.

Final Answer: (i) $\dfrac16$; (ii) $\dfrac12$.

Takeaway: Count the favourable outcomes carefully.

Example 3: A number greater than 4

A die is thrown. Find the probability of a number greater than 4.

Solution:

1. Numbers greater than 4: $\{5,6\}$, so $2$ favourable.
2. $P = \dfrac26 = \dfrac13$.

Final Answer: $\dfrac13$.

Takeaway: "Greater than 4" does not include 4 itself.

Example 4: Elementary events sum to 1

A die is thrown. Verify that the probabilities of the six elementary events add to 1.

Solution:

1. Each face has probability $\dfrac16$.
2. Sum $= 6\times\dfrac16 = 1$.

Final Answer: The total is 1.

Takeaway: All elementary events together are certain, so their probabilities sum to 1.