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Probabillity and Statistics – Probability

📘 Notes

Probability

Probability tells us how likely an event is to happen. It is used to describe chance in situations such as tossing a coin, rolling a dice, or picking a coloured counter from a bag.

A probability can be written as a fraction, a decimal, or a percentage. The value is always between \(0\) and \(1\).

Key facts and formulae

1. Probability scale

\[ 0 \leq \text{probability} \leq 1 \]
Probability Meaning
\(0\) Impossible
\(\dfrac{1}{2}\) Even chance
\(1\) Certain

2. Formula for probability

If outcomes are equally likely, then

\[ \text{Probability}=\frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}. \]

3. Complement rule

The probability that an event does not happen is

\[ P(\text{not }A)=1-P(A). \]

4. Writing answers

A probability can be written in different forms.

\[ \frac{1}{4}=0.25=25\% \]

In many questions, fractions are the best form unless the question asks for a decimal or percentage.

Worked examples

Example 1: Rolling a dice

A fair six-sided dice is rolled. Find the probability of getting a \(4\).

Solution

There is \(1\) favourable outcome and \(6\) possible outcomes.

\[ P(4)=\frac{1}{6}. \]
The probability is \[ \frac{1}{6}. \]

Example 2: Coin toss

A fair coin is tossed. Find the probability of getting heads.

Solution

There are \(2\) possible outcomes: heads and tails.

\[ P(\text{heads})=\frac{1}{2}. \]
The probability is \[ \frac{1}{2}. \]

Example 3: Choosing from a bag

A bag contains \(3\) red counters, \(5\) blue counters, and \(2\) green counters. One counter is chosen at random. Find the probability of choosing a blue counter.

Solution

Total number of counters:

\[ 3+5+2=10. \]

Number of blue counters:

\[ 5. \]
\[ P(\text{blue})=\frac{5}{10}=\frac{1}{2}. \]
The probability is \[ \frac{1}{2}. \]

Example 4: Using the complement rule

A fair dice is rolled. Find the probability of not getting a \(6\).

Solution

First find the probability of getting a \(6\):

\[ P(6)=\frac{1}{6}. \]

Now use the complement rule:

\[ P(\text{not }6)=1-\frac{1}{6}=\frac{5}{6}. \]
The probability is \[ \frac{5}{6}. \]

Common mistakes and exam tips

Common mistakes

  • Forgetting to count the total number of possible outcomes correctly.
  • Writing a probability bigger than \(1\) or smaller than \(0\).
  • Not simplifying fractions.
  • Mixing up the event and the number of outcomes.

Exam tips

  • List the possible outcomes if it helps.
  • Use the formula carefully: favourable outcomes over total outcomes.
  • Check that your answer makes sense on the probability scale from \(0\) to \(1\).
  • Leave your final answer as a fraction unless told otherwise.

Summary

\[ \text{Probability}=\frac{\text{favourable outcomes}}{\text{total outcomes}} \] \[ 0 \leq \text{probability} \leq 1 \] \[ P(\text{not }A)=1-P(A) \]

Probability measures chance. A value near \(0\) means unlikely, a value near \(1\) means likely, and a value of \(1\) means certain.