The Mathematics of Maybe: Introduction to Probability

Contents hide

1Randomness and the Probability Scale

2Two Ways to Measure Probability

3Sample Spaces and Events

4Tree Diagrams

Chapter mind map: how it all connects

1 · Randomness and the probability scaleEvery outcome is known, but which one occurs is unpredictable; likelihood runs from 0 to 1

2 · Two ways to measure probabilityExperimental (from real trials) and theoretical (from equally-likely reasoning)

Introduction to Probability

3 · Sample spaces and eventsEvery possible outcome, and the particular outcomes we care about

4 · Tree diagramsMapping out every outcome of a multi-step experiment, branch by branch

What you will learn in this chapter

What makes an event random, and how probability measures likelihood on a scale of 0 to 1
The difference between experimental and theoretical probability, and how to calculate each
How to write the sample space of an experiment and identify an event within it
How to use a tree diagram to find the probability of outcomes in a multi-step experiment

randomnessprobability scaleexperimental probabilitytheoretical probabilitysample spaceeventtree diagram

1Randomness and the Probability Scale

Learn by heartDefinition 1

A random experiment is a repeatable action (tossing a coin, rolling a die) where every possible result is known in advance, but which result occurs on any single try cannot be predicted.

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). Every event’s probability satisfies $0 \le P(E) \le 1$ .

Impossible

Less likely

Even chance

More likely

1 Certain

Event	Where it sits on the scale
Getting a number greater than 6 on a standard die	Impossible (0)
Rolling a 3 on a standard die	Less likely
Flipping a coin and getting heads	Even chance (0.5)
Drawing a red sweet from a bag of all red sweets	Certain (1)

Did you know?

Snakes and Ladders evolved from an ancient Indian dice game, Jñān-Chaupaḍ, where each ladder stood for a virtue and each snake for a vice, teaching moral lessons through the randomness of the dice.

2Two Ways to Measure Probability

Formula

Experimental Probability

= \dfrac{\text{Number of times the event occurred}}{\text{Total number of trials}}

Also called the relative frequency: based on real data from actually performing the experiment

Given

A die is rolled 50 times; it lands on 4 exactly 8 times

Find

Experimental probability of rolling a 4

Solution

$P(4) = \dfrac{8}{50} = 0.16 = \;$ $16\%$

Formula

Theoretical Probability

P(A) = \dfrac{\text{Number of favourable outcomes}}{\text{Number of possible outcomes}}

Assumes every outcome is equally likely; no experiment needed

Given

A letter is picked at random from the word “PROBABILITY”

Find

Probability of picking the letter B

Solution

PROBABILITY has 11 letters, of which 2 are B. $P(B) = \dfrac{2}{11} \approx \;$ $0.182$ or $18.2\%$

The Law of Large Numbers

Experimental probability can differ from theoretical probability, especially with few trials. As the number of trials grows, experimental probability gets closer and closer to theoretical probability.

Gambler’s Fallacy

If a fair coin lands heads six times in a row, many people believe tails is “due” next. This is false: each toss is independent. The probability of tails on the next toss is still exactly $\tfrac12$ , no matter what happened before. The coin has no memory.

3Sample Spaces and Events

Learn by heartDefinition 2

The sample space $S$ is the list of every possible outcome of a random experiment, each listed exactly once. The number of outcomes is the sample size, $n(S)$ . An event is any subset of the sample space, i.e. any one or more of the possible outcomes we’re interested in.

Experiment	Sample space $S$	$n(S)$
Tossing one coin	{H, T}	2
Rolling one die	{1, 2, 3, 4, 5, 6}	6
Tossing two coins	{HH, HT, TH, TT}	4

Solved Example

Q. For rolling one die, write the event “the number rolled is greater than 4”.

$S = \{1,2,3,4,5,6\}$ . The event is $E = \;$ $\{5, 6\}$

Exam Tip

Make sure the sample space matches the detail the question needs. “Will it rain?” might need only {Rain, No Rain}, but a question about rainfall amount needs {No Rain, Drizzle, Light Rain, Heavy Rain} instead.

4Tree Diagrams

A tree diagram maps out every outcome of a multi-step experiment (several independent trials one after another), branch by branch.

Figure 1: Tossing a fair coin twice. Four branches, each equally likely, giving sample space {HH, HT, TH, TT}. — **Figure 1**: Tossing a fair coin twice. Four branches, each equally likely, giving sample space {HH, HT, TH, TT}.

Solved Example

Q. Using the tree diagram, find the probability of getting exactly one head and one tail in two tosses.

Outcomes with exactly one H and one T: HT and TH, so 2 out of 4 outcomes.

$P(\text{one H, one T}) = \dfrac{2}{4} = \;$ $0.5$ or $50\%$

Probability never tells you what will happen next; it tells you what will happen on average, over many, many tries.

the single idea to carry out of this chapter

Check yourself before the exam

Can I place an event correctly on the 0-to-1 probability scale?
Can I calculate both experimental and theoretical probability, and explain the difference?
Can I write out the sample space of an experiment and identify an event within it?
Can I draw a tree diagram for a two-step experiment and use it to find a probability?

Quick Revision: read this the night before the exam

$0 \le P(E) \le 1$ ; 0 = impossible, 1 = certain
Experimental probability $=\frac{\text{times event occurred}}{\text{total trials}}$ ; theoretical $=\frac{\text{favourable outcomes}}{\text{possible outcomes}}$
Law of Large Numbers: more trials brings experimental probability closer to theoretical
Gambler’s Fallacy: independent trials never “remember” past results
Sample space $S$ = every outcome, listed once; event = a subset of $S$
Tree diagrams list every outcome of a multi-step experiment, branch by branch

Practice Questions

1 markWhat is the probability of an event that is certain to happen?
2 marksIn a survey of 60 students, 18 said cricket was their favourite sport. Find the relative frequency of students who like cricket.
2 marksA bag has 5 red, 3 blue and 2 green balls. One ball is picked at random. Find the probability it is not red.
2 marksTwo coins are tossed together. What is the probability of getting at least one head?
2 marksWrite the sample space for rolling a die and tossing a coin together, and find $n(S)$ .
2 marksThe letters of the word “STATISTICS” are placed on cards, and one is drawn at random. Find the probability it is a “T”.
3 marksA spinner has 8 equally likely numbers, 1 to 8. Find the probability of it landing on: (i) a multiple of 3, (ii) an odd number.
3 marksDraw a tree diagram for tossing a coin and then rolling a die. List the sample space and find $n(S)$ .
2 marksA die is rolled 40 times, landing on 6 exactly 5 times. Find the experimental probability of rolling a 6.
3 marksExplain, with an example, why the Gambler’s Fallacy is a mistaken way of reasoning about probability.

Multiple Choice

1 markThe probability of an impossible event is:
(a) 1(b) 0.5(c) 0(d) undefined
1 markTheoretical probability assumes:
(a) many trials have been performed(b) all outcomes are equally likely(c) the event is impossible(d) nothing about the outcomes
1 markFor tossing 3 coins, the sample size $n(S)$ is:
(a) 3(b) 6(c) 8(d) 9

Assertion (A): If a fair coin has landed heads five times in a row, the next toss is more likely to be tails.
Reason (R): Each coin toss is an independent event, and the coin has no memory of previous tosses.

Case Study

A tyre company tested 1000 tyres and recorded the distance each travelled before needing replacement: 20 lasted less than 4000 km, 210 lasted 4001-9000 km, 325 lasted 9001-14000 km, and 445 lasted more than 14000 km.

1 markWhat is the probability a randomly chosen tyre lasts less than 4000 km?
1 markWhat is the probability it lasts more than 14000 km?
2 marksWhat is the probability it lasts between 4000 and 14000 km (inclusive of both groups)?

Answer Key

Q11

18/60 = 0.3

30\%

Q3Not red

=5

out of

10

P=1-\frac{5}{10}=0.5

S=\{HH,HT,TH,TT\}

; at least one head

=3/4=0.75

S=\{(1,H),(1,T),(2,H),\ldots,(6,T)\}

;

n(S)=12

Q6STATISTICS has 10 letters, 3 T’s:

P(T)=3/10=0.3

Q7(i) multiples of 3 in 1-8: {3,6},

P=2/8=0.25

; (ii) odd: {1,3,5,7},

P=4/8=0.5

S=\{(H,1),(H,2),\ldots,(T,6)\}

;

n(S)=12

P(6)=5/40=0.125

12.5\%

Q10Each toss/roll is independent; past outcomes never change the fixed probability of future ones, even after a long streak

MCQ 1–3(c) · (b) · (c)

A/RA is false, R is true (Gambler’s Fallacy; the probability stays 0.5 regardless of past tosses)

Case Study

20/1000=0.02

445/1000=0.445

(210+325)/1000=0.535