Basic Probability

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Notes

Basic Probability

Probability is a number between **0** (impossible) and **1** (certain).
Probability scale: $0 =$ impossible, $0.5 =$ even chance, $1 =$ certain.
**P(A)** $= (number$ of outcomes in event A) / (total number of equally likely outcomes).
All probabilities sum to **1**.
**Complement**: $P(not A) = 1$ – P(A).
**Mutually exclusive** events cannot happen together; P(A or $B) = P(A) + P(B)$ .
Probabilities can be fractions, decimals, or percentages.

Possibility (Sample Space) Diagrams

A **sample space** lists all possible outcomes of an experiment.
For two events (e.g., rolling two dice), use a **grid** (possibility diagram).
Each cell in the grid represents one equally likely outcome.
Probability $= (number$ of desired outcomes) / (total outcomes in sample space).
Counting method works only if all outcomes are **equally likely**.
For three or more events, list outcomes systematically (e.g., HHH, HHT, ...).

Relative Frequency

**Relative frequency** $= (number$ of successful trials) / (total number of trials).
It estimates probability when theoretical probability is unknown.
More trials give a more accurate estimate (law of large numbers).
Compare relative frequency to theoretical probability to test fairness/bias.
Trials must be **independent** and **random** (e.g., replace items).

Expected Frequency

**Expected frequency** = probability × number of trials.
It predicts how many times an outcome will occur in a given number of trials.
Example: $P(red) = 0.2, 500$ trials → expected reds $= 0.2 \times 500 = 100$ .
If relative frequency is known, use it as the probability in the formula.
The best estimate comes from the experiment with the **most trials**.

Worked Examples

**Example 1**: Bag with 6 blue, 4 red, 5 yellow. $P(yellow) = \frac{5}{15} = \frac{1}{3}$ . Expected in 300 draws: $300 \times \frac{1}{3} = 100$ .
**Example 2**: Biased coin flipped 40 times → 10 heads. Relative frequency $= \frac{10}{40} = 0.25$ . Expected heads in 100 flips: $0.25 \times 100 = 25$ .
**Example 3**: Two dice rolled. $P(sum > 5$ and $odd) = \frac{12}{36} = \frac{1}{3}$ . Given that, P(one die shows $2) = \frac{2}{12} = \frac{1}{6}$ .

Probability Scale

Possibility Diagram for Two Dice Sum

Relative Frequency vs Theoretical Probability

Expected Frequency Formula

Practice questions

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1.The probability that a sweet made in a factory is the wrong shape is 0.0028. One day, the factory makes 25 000 sweets. Calculate the number of sweets that are expected to be the wrong shape.
Easy
- A70
- B700
- C7
- D0.7
2.The probability that Kim wins a game is 0.72. In one year Kim will play 225 games. Work out an estimate of the number of games Kim will win.
Easy
- A162
- B1620
- C16.2
- D324
3.Sushila has a bag which contains 10 red balls and 8 blue balls. Sushila takes one ball at random from her bag. Find the probability that she takes a red ball.
Easy
- A $\frac{10}{18}$
- B $\frac{8}{18}$
- C $\frac{10}{8}$
- D $\frac{1}{2}$
4.The time taken for each of 120 students to complete a cooking challenge is shown in the table. Time $(t minutes): 20<t\le 25 (44), 25<t\le 30 (32), 30<t\le 35 (28), 35<t\le 40 (12), 40<t\le 45 (4)$ . A student is chosen at random. Find the probability that this student takes more than 40 minutes.
Easy
- A $\frac{4}{120}$
- B $\frac{12}{120}$
- C $\frac{28}{120}$
- D $\frac{44}{120}$
5.The frequency table shows information about the time, m minutes, that each of 160 people spend in a library. Time $(m minutes): 0<m\le 10 (3), 10<m\le 40 (39), 40<m\le 60 (43), 60<m\le 90 (55), 90<m\le 100 (11), 100<m\le 120 (9)$ . Find the probability that one of these people, chosen at random, spends more than 100 minutes in the library.
Easy
- A $\frac{9}{160}$
- B $\frac{11}{160}$
- C $\frac{20}{160}$
- D $\frac{55}{160}$
6.On any given day the probability that it is sunny is 2/5. In a period of 90 days, on how many days is it expected to be sunny?
Easy
- A36
- B54
- C45
- D18
7.Sofia has a bag containing 8 blue beads and 7 red beads only. She takes one bead out of the bag at random and replaces it. She does this 90 times. Find the number of times she expects to take a red bead.
Medium
- A42
- B48
- C45
- D35
8.One of the teachers at a school is chosen at random. The probability that this teacher is female is 3/5. There are 36 male teachers at the school. Work out the total number of teachers at the school.
Medium
- A90
- B60
- C54
- D72

Summary

## Basic Probability ### Probability Scale - 0 = impossible, 1 = certain. - Can be fraction, decimal, or percentage. ### Key Definitions - **Experiment**: repeated activity; each repeat is a **trial**. - **Outcome**: possible result of a trial. - **Event**: an outcome or set of outcomes (e.g., rolling an even number). - **Sample space**: all possible outcomes. - **Fair**: all outcomes equally likely; **biased**: not equally likely. ### Calculating Probability If all outcomes equally likely: \[ P(A) = \frac{\text{number of outcomes for event }A}{\text{total number of outcomes}} \] ### Missing Probabilities - All probabilities sum to 1. - **Complement**: $ P(\text{not }A) = 1 - P(A) $. ### Mutually Exclusive Events - Cannot happen at the same time. - $ P(A \text{ or } B) = P(A) + P(B) $. ### Possibility (Sample Space) Diagrams - Grid for two sets of outcomes (e.g., two dice). - Count desired outcomes; divide by total. ### Relative Frequency - Estimate from experiment: $ \frac{\text{number of successful outcomes}}{\text{total number of trials}} $. - More trials → more accurate. - Used when theoretical probability unknown. ### Expected Frequency - $ \text{Expected frequency} = \text{probability} \times \text{number of trials} $. - Also: $ \text{relative frequency} \times \text{number of trials} $. ### Tips - Use fractions unless told otherwise. - For best estimate, choose experiment with most trials. - Replace items in bag for independent trials.

Key terms

Probability: A measure of how likely an event is to occur, ranging from 0 (impossible) to 1 (certain).
Sample space: The set of all possible outcomes of an experiment.
Event: An outcome or a collection of outcomes.
Mutually exclusive: Events that cannot happen at the same time.
Relative frequency: An estimate of probability based on experimental data: number of successful outcomes divided by total number of trials.
Expected frequency: The number of times an outcome is predicted to occur: probability multiplied by number of trials.
Complement: The event that an event does not happen; P(not A) = 1 - P(A).
Possibility diagram: A grid showing all possible outcomes of two events (e.g., rolling two dice).
Fair: When all outcomes have an equal chance of occurring.
Biased: When outcomes do not have equal chances; the probability is not uniform.

Slides

What is Probability?

Probability measures how likely something is to happen, from 0 (impossible) to 1 (certain).
It can be written as a fraction, decimal, or percentage.
Key terms: experiment (activity repeated), trial (each repeat), outcome (possible result), event (outcome or set of outcomes).

Which of the following best describes a probability of 0.5?

Impossible
Unlikely
Even chance
Certain

Calculating Basic Probability

If all outcomes are equally likely, $P(event) = (number$ of ways event can happen) / (total number of outcomes).
Example: A bag has 10 red and 8 blue balls. $P(red) = 10/(10+8) = \frac{10}{18} = \frac{5}{9}$ .
Probabilities of all outcomes add up to 1.

A bag contains 3 blue, 4 red, and 8 green pens. The probability of picking a green pen $is ____$ .

Complementary Events

The complement of event A is 'not A'.
$P(not A) = 1 - P(A)$ .
Example: If probability of rain is 0.3, probability of no rain is 0.7.

The probability that a seed grows is 0.75. What is the probability it does not grow?

0.25
0.75
0.5
1

Mutually Exclusive Events

Events are mutually exclusive if they cannot happen at the same time.
For mutually exclusive events A and B, P(A or $B) = P(A) + P(B)$ .
Example: Rolling a dice, 'getting a 2' and 'getting a 5' are mutually exclusive.

Which of these are true?

If A and B are mutually exclusive, P(A or $B) = P(A) + P(B)$ .
Complementary events are always mutually exclusive.
Mutually exclusive events can happen at the same time.

Possibility (Sample Space) Diagrams

A sample space lists all possible outcomes of an experiment.
For two events (e.g., rolling two dice), a grid (possibility diagram) is useful.
Probabilities are found by counting desired outcomes over total outcomes.

When rolling two fair dice, the total number of outcomes $is ____$ .

Relative Frequency

Relative frequency estimates probability from experiments: (number of successful trials) / (total trials).
It gets more accurate with more trials.
Used when theoretical probability is unknown.

A coin is flipped 50 times and lands heads 20 times. What is the relative frequency of heads?

$\frac{20}{50}$
$\frac{30}{50}$
$\frac{20}{30}$
$\frac{50}{20}$

Expected Frequency

Expected frequency = probability × number of trials.
Example: If $P(win) = 0.72$ and 225 games are played, expected wins $= 0.72 \times 225 = 162$ .
It gives an estimate of how many times an outcome will occur.

The probability of a sweet being wrong shape is 0.0028. In 25000 sweets, the expected number of wrong shape sweets $is ____$ .

Using Relative Frequency to Test Fairness

Compare relative frequency to theoretical probability to check if an event is fair or biased.
If relative frequency is close to theoretical probability, it's likely fair.
If not, the event may be biased.

A biased coin is flipped 100 times and lands heads 13 times. Which is true?

The coin is likely fair
The coin is likely biased
The theoretical probability is 0.13
Relative frequency is $\frac{13}{100}$

Summary

Probability ranges from 0 to 1.
$P(event) =$ number of ways event can happen / total outcomes.
Complement: $P(not A) = 1 - P(A)$ .
Mutually exclusive: P(A or $B) = P(A) + P(B)$ .
Relative frequency estimates probability from experiments.
Expected frequency = probability × number of trials.

Which of these are true?

Expected frequency is always exact.
Relative frequency improves with more trials.
The sum of probabilities of all outcomes is 1.

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