Basic Probability

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Notes

Basic Probability Concepts

Probability is a number between 0 and 1 describing likelihood: $0 =$ impossible, $1 =$ certain.
An **experiment** is an activity repeated to produce results; each repeat is a **trial**.
An **outcome** is a possible result of a trial; an **event** is an outcome or collection of outcomes.
The **sample space** is the set of all possible outcomes.
If all outcomes are equally likely, the probability of an event A is $P(A) = (number$ of outcomes in A) / (total number of outcomes).
Probabilities can be given as fractions, decimals, or percentages.

Probability Scale

The probability scale runs from 0 to 1.
$0 =$ impossible, between 0 and $0.5 =$ unlikely, $0.5 =$ even chance, between 0.5 and $1 =$ likely, $1 =$ certain.
Probabilities of all outcomes in a sample space sum to 1.
The complement of event A (not A) has probability $P(not A) = 1 - P(A)$ .

Mutually Exclusive Events

Two events are **mutually exclusive** if they cannot happen at the same time.
For mutually exclusive events A and B, P(A or $B) = P(A) + P(B)$ .
Complementary events are always mutually exclusive.

Calculating Basic Probabilities

If all outcomes equally likely, probability of a specific outcome $= 1 /$ total number of outcomes.
Example: A bag with 50 marbles, 20 blue → $P(blue) = \frac{20}{50} = \frac{2}{5}$ .
To find a missing probability in a table, make all probabilities sum to 1.
If two probabilities are equal and their sum is known, divide by 2 to find each.

Possibility (Sample Space) Diagrams

A **possibility diagram** (or sample space diagram) is a grid showing all outcomes when combining two sets of outcomes.
For example, rolling two dice: grid shows sums from 2 to 12, with 36 equally likely outcomes.
Probabilities are found by counting favourable outcomes and dividing by total outcomes.
This method only works if all outcomes in the sample space are equally likely.

Relative Frequency

**Relative frequency** $= (number$ of successful trials) / (total number of trials).
It is an estimate of probability based on experiment, used when theoretical probability is unknown.
The more trials, the more accurate the estimate becomes.
Relative frequency can be compared to theoretical probability to test fairness or bias.

Expected Frequency

**Expected frequency** = probability × number of trials.
It gives the number of times you would expect an outcome to occur in a given number of trials.
Example: If $P(yellow) = \frac{1}{3}$ and 300 trials, expected yellow $= 300 \times \frac{1}{3} = 100$ .
Expected frequency can be based on theoretical probability or relative frequency.

Key Terminology

**Fair** means all outcomes equally likely; **biased** means not equally likely.
n(A) denotes the number of outcomes in event A.
P(A) denotes the probability of event A.
Trials must be independent and random for relative frequency to be valid.

Probability Scale

Possibility Diagram for Sum of Two Dice

Relative Frequency Formula

Expected Frequency Formula

Practice questions

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1.A fair spinner has the numbers 2, 2, 3, 4, 4, 5. What is the probability that it lands on an even number?
Easy
- A $\frac{1}{2}$
- B $\frac{2}{3}$
- C $\frac{1}{3}$
- D $\frac{5}{6}$
2.A bag contains 3 green, 4 red, and 1 blue ball. What is the probability of taking a red ball?
Easy
- A $\frac{4}{8}$
- B $\frac{4}{7}$
- C $\frac{1}{2}$
- D $\frac{3}{8}$
3.The probability that it rains tomorrow is 0.35. What is the probability that it does not rain tomorrow?
Easy
- A0.65
- B0.35
- C0.5
- D0.75
4.A box contains 22 pencils: 6 pink, 9 blue, 7 yellow. What is the probability of taking a green pencil?
Easy
- A0
- B $\frac{1}{22}$
- C $\frac{7}{22}$
- D $\frac{6}{22}$
5.The probability that Alex wins a prize is 0.27. What is the probability that Alex does not win a prize?
Easy
- A0.73
- B0.27
- C0.5
- D0.63
6.A bag contains 6 red and 10 blue balls. What is the probability of taking a blue ball?
Easy
- A $\frac{10}{16}$
- B $\frac{6}{16}$
- C $\frac{10}{10}$
- D $\frac{1}{2}$
7.A bag contains 20 balls, 5 of which are red. A ball is picked at random. On a probability scale from 0 to 1, which arrow shows the probability that the ball is red?
Medium
- A0.25
- B0.5
- C0.75
- D1
8.A spinner has 8 sides numbered 3, 4, 4, 7, 7, 7, 8, 9. What is the probability of landing on 7?
Medium
- A $\frac{3}{8}$
- B $\frac{1}{8}$
- C $\frac{1}{3}$
- D $\frac{3}{7}$

Summary

## Basic Probability - **Probability scale**: 0 (impossible) to 1 (certain). - **Key formula**: $ P(A) = \frac{\text{number of outcomes for event } A}{\text{total number of outcomes}} $ (equally likely outcomes). - **Complement**: $ P(\text{not } A) = 1 - P(A) $. - **Mutually exclusive events**: $ P(A \text{ or } B) = P(A) + P(B) $. - **All probabilities sum to 1**. ## Possibility (Sample Space) Diagrams - List or grid showing all possible outcomes. - Count desired outcomes and divide by total. ## Relative & Expected Frequency - **Relative frequency**: $ \frac{\text{number of successful trials}}{\text{total number of trials}} $. - **Expected frequency**: $ \text{probability} \times \text{number of trials} $. - More trials → better estimate. **Tips**: Use fractions unless told otherwise. For "not" questions, use complement. For "or" with mutually exclusive, add probabilities.

Key terms

Probability: A number between 0 and 1 that describes the likelihood of an event occurring.
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.
Complement: The event that the original event does not occur; P(not A) = 1 - P(A).
Relative frequency: An estimate of probability based on experimental data: number of successful trials divided by total trials.
Expected frequency: The number of times an outcome is expected to occur: probability × number of trials.

Slides

What is Probability?

Probability describes the likelihood of something happening.
It is measured on a scale from 0 (impossible) to 1 (certain).
Probabilities can be written as fractions, decimals or percentages.

What is the probability of an impossible event?

0
0.5
1
Cannot be determined

Calculating Basic Probability

If all outcomes are equally likely, probability of an event $= (number$ of outcomes of the event) / (total number of outcomes).
For example, if a bag has 3 red and 2 blue marbles, $P(red) = \frac{3}{5}$ .
The sum of probabilities of all outcomes is 1.

A bag contains 4 green balls and 6 red balls. The probability of picking a green ball $is ____$ .

Complementary Events

The complement of event A is 'not A'.
$P(not A) = 1 - P(A)$ .
For example, if $P(rain) = 0.3$ , then $P(no rain) = 0.7$ .

The probability that it rains tomorrow is 0.35. What is the probability it does not rain?

0.35
0.65
0.7
1

Mutually Exclusive Events

Events are mutually exclusive if they cannot happen at the same time.
For mutually exclusive events, P(A or $B) = P(A) + P(B)$ .
Complementary events are mutually exclusive.

Which of these are true?

If A and B are mutually exclusive, P(A or $B) = P(A) + P(B)$ .
Mutually exclusive events can happen together.
Rolling a 2 and rolling an odd number on a dice are mutually exclusive.

Possibility (Sample Space) Diagrams

A sample space lists all possible outcomes of an experiment.
For two events, a grid (possibility diagram) can show all outcomes.
Probabilities are found by counting desired outcomes over total outcomes.

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

Relative Frequency

Relative frequency estimates probability from experiments: (successful trials)/(total trials).
The more trials, the more accurate the estimate.
It is used when theoretical probability is unknown.

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

$\frac{20}{50}$
$\frac{30}{50}$
$\frac{1}{2}$
$\frac{20}{100}$

Expected Frequency

Expected frequency = probability × number of trials.
It tells how many times you expect an outcome to occur.
For example, if $P(red) = 0.4$ and you draw 100 times, expect 40 reds.

A spinner has $P(blue) = 0.2$ . If spun 300 times, expected number of blues $is ____$ .

Using Probability Tables

Probabilities in a table always add up to 1.
If one probability is missing, subtract the sum of known ones from 1.
For complementary events, $P(not A) = 1 - P(A)$ .

Which of these are true?

The sum of all probabilities in a table is 1.
If $P(red)=0.3, P(blue)=0.5$ , then P(green) must be 0.2.
$P(not A) = P(A) - 1$ .

Key Terms

Experiment: an activity with repeated trials.
Outcome: a possible result of a trial.
Event: a set of outcomes (e.g., rolling an even number).

What is the sample space for rolling a fair six-sided dice?

{1,2,3,4,5,6}
{1,2,3,4,5}
{0,1,2,3,4,5,6}
{Heads, Tails}

Flashcards

What is the probability scale?

A scale from 0 to 1 where 0 means impossible, 1 means certain, 0.5 means even chance.

How do you calculate basic probability?

$P(A) = (number$ of outcomes for event A) / (total number of equally likely outcomes).

What is the complement rule?

P(event does not $happen) = 1 - P(event happens)$ .

What are mutually exclusive events?

Events that cannot happen at the same time. For mutually exclusive events, P(A or $B) = P(A) + P(B)$ .

What is a sample space?

The set of all possible outcomes of an experiment. It can be listed or shown in a table.

What is a possibility diagram?

A grid showing all possible outcomes when combining two sets of outcomes, e.g., rolling two dice.

How do you find probability from a possibility diagram?

Count the number of desired outcomes and divide by the total number of outcomes in the diagram.

What is relative frequency?

An estimate of probability based on experiment: (number of successful trials) / (total number of trials).

What is expected frequency?

The number of times you expect an outcome to occur: probability × number of trials.

A fair spinner has numbers 2,2,3,4,4,5. Find P(even number).

There are 4 even numbers (2,2,4,4) out of 6. $P(even) = \frac{4}{6} = \frac{2}{3}$ .

A bag has 1,1,2,3,3,3,4,5. Find P(odd number).

Odd numbers: 1,1,3,3,3,5 (6 odds) out of 8. $P(odd) = \frac{6}{8} = \frac{3}{4}$ .

A five-sided spinner numbered 2,3,4,5,6. Find P(prime number).

Prime numbers: 2,3,5 (3 primes) out of 5. $P(prime) = \frac{3}{5}$ .

30 flags: 11 pink, 7 yellow, 5 blue, 4 silver, 3 green. Find P(not blue).

Not blue = total - blue $= 30 - 5 = 25$ . $P(not blue) = \frac{25}{30} = \frac{5}{6}$ .

A bag has 3 green, 4 red, 1 blue. Find P(red).

4 red out of 8 total. $P(red) = \frac{4}{8} = \frac{1}{2}$ .

A bag has 6 red and 10 blue balls. Find P(blue).

10 blue out of 16 total. $P(blue) = \frac{10}{16} = \frac{5}{8}$ .

$P(green) = 0.38$ . Find P(red) if only green and red.

$P(red) = 1 - 0.38 = 0.62$ .

$P(win) = 0.27$ . Find P(not win).

$P(not win) = 1 - 0.27 = 0.73$ .

$P(sunny) = 0.97$ . Find P(not sunny).

$P(not sunny) = 1 - 0.97 = 0.03$ .

$P(rain) = 0.35$ . Find P(not rain).

$P(not rain) = 1 - 0.35 = 0.65$ .

A bag has 20 balls, 5 red. Find P(red).

5 red out of 20. $P(red) = \frac{5}{20} = \frac{1}{4}$ .

$P(cycle) = \frac{1}{5}$ . Find P(walk) if only walk or cycle.

$P(walk) = 1 - \frac{1}{5} = \frac{4}{5}$ .

Expected number of days Dan cycles in 200 days if $P(cycle)=\frac{1}{5}$ .

Expected $= 200 \times \frac{1}{5} = 40$ days.

A spinner has numbers 3,4,4,7,7,7,8,9. Find P(7).

Three 7s out of 8. $P(7) = \frac{3}{8}$ .

Spinner from previous card spun 160 times. Expected number of 7s.

Expected $= 160 \times \frac{3}{8} = 60$ .

$P(wrong shape)=0.0028, 25000$ sweets made. Expected wrong shape?

Expected $= 25000 \times 0.0028 = 70$ .

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