Probability

Probability of Events

Year 10 · Year 11

✓By the end of this lesson students will be able to calculate theoretical probabilities for single and combined events.
✓By the end of this lesson students will be able to calculate experimental probabilities (relative frequencies) from given data.
✓By the end of this lesson students will be able to construct and interpret sample space diagrams to list all possible outcomes of an event.
✓By the end of this lesson students will be able to construct and interpret frequency trees to represent successive events.
✓By the end of this lesson students will be able to compare theoretical and experimental probabilities.

Key concepts

Theoretical Probability

Theoretical probability is the likelihood of an event occurring based on reasoning and assumptions about equally likely outcomes, without conducting any experiments. It is what we expect to happen in an ideal situation.

P(Event A) = (Number of outcomes favourable to A) / (Total number of possible outcomes)

Experimental Probability (Relative Frequency)

Experimental probability, also known as relative frequency, is the likelihood of an event occurring based on the results of an experiment or observation. It is calculated from actual trials and may differ from theoretical probability, especially with a small number of trials. As the number of trials increases, the experimental probability tends to get closer to the theoretical probability.

P(Event A) = (Number of times Event A occurs) / (Total number of trials)

Sample Space

A sample space is a list or diagram showing all possible outcomes of an experiment. It helps to systematically identify every possible result, which is crucial for calculating probabilities. Sample spaces can be represented as lists, tables, or tree diagrams.

Frequency Trees

A frequency tree is a diagram used to represent the frequencies of two or more successive events. It starts with a total frequency, which then branches out to show the frequencies of different outcomes for the first event, and then further branches out for the second event. Each branch is labelled with the frequency of that particular outcome.

Key facts to remember

1Probability is always a value between 0 and 1, inclusive. P=0 means an event is impossible, P=1 means an event is certain.
2The sum of probabilities of all possible outcomes in a sample space is always 1.
3P(Event A) + P(Not Event A) = 1. This is known as the complement rule.
4Theoretical probability relies on assumptions of fairness and equally likely outcomes.
5Experimental probability (relative frequency) is based on observed results from trials and can vary with the number of trials.
6A sample space lists all possible outcomes of an experiment.
7Frequency trees are useful for organising and calculating frequencies of successive events.
8Probabilities are usually expressed as fractions, decimals, or percentages.

Worked examples

Example 1

A fair six-sided dice is rolled 80 times. The results are recorded in the table below: | Score | Frequency | |-------|-----------| | 1 | 12 | | 2 | 15 | | 3 | 10 | | 4 | 18 | | 5 | 13 | | 6 | 12 | a) What is the theoretical probability of rolling a 4? b) What is the experimental probability of rolling a 4? c) Compare the theoretical and experimental probabilities for rolling a 4.

Ia) For a fair six-sided dice, there are 6 equally likely outcomes (1, 2, 3, 4, 5, 6). Only one of these outcomes is a 4.

II Theoretical P(rolling a 4) = (Number of favourable outcomes) / (Total number of possible outcomes) = 1/6

IIIb) From the table, the score 4 occurred 18 times out of a total of 80 rolls.

IV Experimental P(rolling a 4) = (Number of times 4 occurred) / (Total number of trials) = 18/80

V Simplify the fraction: 18/80 = 9/40

VIc) To compare, convert both probabilities to decimals or percentages (optional, but often helpful for comparison).

VII Theoretical P(rolling a 4) = 1/6 ≈ 0.1667 (or 16.67%)

VIII Experimental P(rolling a 4) = 9/40 = 0.225 (or 22.5%)

9 The experimental probability (0.225) is higher than the theoretical probability (0.1667) in this set of trials. This difference is due to the random nature of experimental results over a finite number of trials.

Answer

a) 1/6 b) 9/40 c) The experimental probability (9/40) is higher than the theoretical probability (1/6).

Always simplify fractions in your final answer unless otherwise specified.

Example 2

Two fair coins are tossed. a) Draw a sample space diagram to show all possible outcomes. b) Find the probability of getting exactly one head. c) Find the probability of getting at least one tail.

Ia) Let H represent a Head and T represent a Tail. We can use a table or a list to show the sample space.

II Coin 1 | Coin 2 | Outcome

III -------|--------|--------

IV H | H | HH

V H | T | HT

VI T | H | TH

VII T | T | TT

VIII The sample space is {HH, HT, TH, TT}. There are 4 possible outcomes.

9b) We need to find outcomes with exactly one head. From the sample space, these are HT and TH.

10 There are 2 favourable outcomes.

11 P(exactly one head) = (Number of favourable outcomes) / (Total number of possible outcomes) = 2/4 = 1/2

12c) We need to find outcomes with at least one tail. This means one tail or two tails. From the sample space, these are HT, TH, and TT.

13 There are 3 favourable outcomes.

14 P(at least one tail) = (Number of favourable outcomes) / (Total number of possible outcomes) = 3/4

Answer

a) Sample space: {HH, HT, TH, TT} b) 1/2 c) 3/4

Ensure your sample space includes all unique, equally likely outcomes.

Example 3

150 students were asked whether they preferred tea or coffee. 90 of the students were girls. 45 of the boys preferred coffee. 30 of the girls preferred tea. a) Complete a frequency tree to represent this information. b) What is the probability that a randomly chosen student is a boy who prefers tea?

Ia) Start with the total number of students and branch out for gender, then for drink preference.

II Total Students: 150

III Branch 1 (Gender):

IV Girls: 90

V Boys: 150 - 90 = 60

VI Branch 2 (Drink Preference for Girls):

VII Girls who prefer Tea: 30 (given)

VIII Girls who prefer Coffee: 90 - 30 = 60

9 Branch 2 (Drink Preference for Boys):

10 Boys who prefer Coffee: 45 (given)

11 Boys who prefer Tea: 60 - 45 = 15

12 (Visualise or sketch the tree as you fill these in)

13b) From the completed frequency tree, the number of boys who prefer tea is 15.

14 The total number of students is 150.

15 P(Boy and prefers Tea) = (Number of boys who prefer tea) / (Total number of students) = 15/150

16 Simplify the fraction: 15/150 = 1/10

Answer

a) Frequency tree: Total (150) ├── Girls (90) │ ├── Tea (30) │ └── Coffee (60) └── Boys (60) ├── Tea (15) └── Coffee (45) b) 1/10

Always check that the numbers on the branches add up correctly to the previous branch or total.

Common mistakes

✗Not simplifying fractions to their lowest terms for final probability answers.
✗Confusing theoretical and experimental probability, or using the wrong method for the question asked.
✗Incorrectly listing the sample space, either by missing outcomes or including duplicates, leading to incorrect total outcomes.
✗Errors in calculating frequencies in frequency trees, such as not ensuring branches add up correctly.
✗Expressing probability as a ratio (e.g., 1:5) instead of a fraction, decimal, or percentage.

Exam tips

★Read the question carefully to determine if theoretical or experimental probability is required.
★Always show your working, especially when calculating probabilities from sample spaces or frequency trees.
★For sample space questions, systematically list all outcomes to avoid missing any. Tables or tree diagrams can be very helpful.
★When constructing frequency trees, fill in all given information first, then use subtraction to find the missing frequencies. Double-check your totals.

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