DP Maths AI HL 4.7 Probability Distributions

DP Maths AI HL 4.7 Probability Distributions - Full notes

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DP Maths AI HL | 4.7 Probability Distributions

DP MATHEMATICS AI · TOPIC 4: STATISTICS AND PROBABILITY

4.7

Probability Distributions

Full notes for Applications and Interpretation HL

Contents

1	Random variables
2	Discrete probability distributions
3	Distributions given by formulas
4	Expected value
5	Fair games and expected gain
6	Expected value in decision making
7	Variance and standard deviation of a discrete variable
8	The binomial distribution
9	Mean and variance of the binomial distribution
10	Continuous variables and the normal distribution
11	Properties of the normal curve
12	Normal probabilities
13	Inverse normal calculations
14	Standardization and z-scores
15	Finding an unknown mean or standard deviation
16	Combining the normal and binomial distributions
17	Checking model assumptions
18	Mixed review: worked problems
19	The Poisson distribution (HL)
20	Linear combinations of random variables (HL)
21	Investigation: insurance premiums
22	Exam-style questions with markscheme
23	Common misconceptions and links
24	Check your understanding
25	Summary and topic map
26	Answers to practice exercises

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and work through every worked example before attempting the practice exercise.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
define discrete random variables and their probability distributions
use ΣP(X = x) = 1 to find unknowns
calculate and interpret expected value, including fair games
use the binomial distribution, its mean and variance
describe the normal distribution and its properties
calculate normal probabilities and inverse normal values
use z-scores and find unknown parameters
combine normal and binomial models in context

HL booklet: this booklet contains all the SL content plus the Poisson distribution and linear combinations of random variables.

Guiding questions

How can the long-run behaviour of random processes be predicted? When is the normal distribution an appropriate model?

How this topic is examined

Probability distributions are central to Paper 2: complete a probability table and find E(X), decide whether a game is fair, use the binomial distribution for repeated trials, and use the normal distribution (and inverse normal) for measurements. Combined questions are common.

Define the random variable in words and state its distribution, for example X ~ B(20, 0.3) or X ~ N(65, 82), before calculating.

Mathematics: Applications and Interpretation places strong emphasis on technology, modelling and interpretation. Most papers allow a GDC, so you are expected to use it efficiently, record the method you used, and interpret every answer in context.

Command term	What the examiner wants
Write down	an answer with little or no working; it should be obtainable by inspection
Find / Calculate / Determine	obtain the answer showing relevant working, including what you entered into a GDC
Show that	obtain the given result; every step must be shown and the result must not be assumed
Estimate	obtain an approximate value
Sketch	a diagram showing the general shape and key features, labelled
Interpret / Comment	explain the meaning of a result in context, or give a judgement
Justify / Explain	give valid reasons or evidence
Hence	use the previous result

How to use these notes

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or explanation of a key result
Key point	a result or idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
GDC tip	how to use your graphic display calculator efficiently
In the real world	where the mathematics is used outside the classroom
Worked example	a full solution with IB-style marks (M1 method, A1 answer, R1 reasoning, AG answer given), graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet

1 Random variables

Numbers from chance.

Think about it

A game pays $5 if a die shows 6, and nothing otherwise. If you played it 600 times, about how much would you expect to win in total?

A random variable X assigns a number to each outcome of a random process. X is discrete if it takes separate (countable) values, such as the number of heads; continuous if it can take any value in an interval, such as a mass or a time.

2 Discrete probability distributions

Values and their probabilities.

A probability distribution lists each value x with P(X = x). The probabilities must each be between 0 and 1 and must sum to 1.

x	0	1	2	3
P(X = x)	0.1	0.3	k	0.2

WORKED EXAMPLE 1 Finding k (basic)

Find k and P(X ≥ 2) for the table.

Solution

0.1 + 0.3 + k + 0.2 = 1: k = 0.4A1

P(X ≥ 2) = 0.6A1

Practice exercise 1

1. X takes values 1, 2, 3, 4 with probabilities 0.15, 0.25, 0.4, p. Find p and P(X < 3).

2. Explain why a table with probabilities 0.5, 0.4, 0.3 is not a valid distribution.

3 Distributions given by formulas

P(X = x) = f(x).

WORKED EXAMPLE 2 Formula (standard)

P(X = x) = kx for x = 1, 2, 3, 4. Find k and P(X = 3).

Solution

k(1 + 2 + 3 + 4) = 1: k = 0.1A1

P(X = 3) = 0.3A1

Practice exercise 2

1. P(X = x) = k(x2 + 1) for x = 0, 1, 2. Find k.

4 Expected value

The long-run mean.

E(X) is the mean value of X over many repetitions. It need not be a possible value: for a fair die, E(X) = 3.5.

WORKED EXAMPLE 3 Expected value (standard)

Find E(X) for the table with k = 0.4.

Solution

0(0.1) + 1(0.3) + 2(0.4) + 3(0.2)M1

= 1.7A1

Practice exercise 3

1. Find E(X) for a fair die.

2. X takes values 2, 5, 10 with probabilities 0.5, 0.3, 0.2. Find E(X).

5 Fair games and expected gain

E(gain) = 0.

Let X be a player's gain (winnings minus stake). The game is fair if E(X) = 0. If E(X) < 0, the player loses on average and the organizer profits.

WORKED EXAMPLE 4 Fair game (exam level)

A game costs $3. A spinner gives $10 with probability 0.15, $4 with probability 0.25 and nothing otherwise. Find the expected gain and whether the game is fair.

Solution

Expected winnings = 1.5 + 1 = $2.50M1 A1

Expected gain = 2.50 − 3 = −$0.50: not fairA1 R1

Practice exercise 4

1. Find the cost that would make the spinner game fair.

2. A raffle sells 1000 tickets at $5 with prizes $1000, $500 and $250. Find the expected gain per ticket.

6 Expected value in decision making

Choosing the best option.

WORKED EXAMPLE 5 Insurance (exam level)

A phone worth $800 has a 6% chance of being lost and a 10% chance of a $200 repair in a year. Insurance costs $90. Should the owner insure?

Solution

Expected cost without insurance = 0.06(800) + 0.10(200) = $68M1 A1

$68 < $90: on average, not insuring is cheaper, though insurance reduces riskR1

Practice exercise 5

1. A farmer can plant crop A (profit $8000 with probability 0.7, loss $2000 otherwise) or crop B (certain profit $5000). Which has higher expected profit?

7 Variance and standard deviation of a discrete variable

Spread of outcomes.

GDC tip

Enter values in L1 and probabilities in L2, and run 1-Var Stats L1, L2: x̄ gives E(X) and σx gives the standard deviation.

WORKED EXAMPLE 6 Variance (standard)

Find Var(X) for the table (E(X) = 1.7).

Solution

E(X2) = 0 + 0.3 + 1.6 + 1.8 = 3.7A1

Var(X) = 3.7 − 2.89 = 0.81, SD 0.9A1

8 The binomial distribution

Successes in fixed independent trials.

X ~ B(n, p) counts successes in n independent trials, each with the same success probability p. Conditions: fixed n; two outcomes; constant p; independence. Use binomial pdf for P(X = r) and binomial cdf for P(X ≤ r). (See the binomial distribution booklet for more detail.)

WORKED EXAMPLE 7 Binomial (standard)

20% of light bulbs are energy-saving. In a box of 12, find P(exactly 3) and P(at least 3).

Solution

X ~ B(12, 0.2): P(X = 3) = 0.236A1

P(X ≥ 3) = 1 − P(X ≤ 2) = 0.442M1 A1

Practice exercise 6

1. X ~ B(10, 0.6). Find P(X = 7) and P(X < 5).

2. State one condition that fails when drawing 5 cards without replacement from 10.

9 Mean and variance of the binomial distribution

np and np(1 − p).

Practice exercise 7

1. Find the mean and SD of B(12, 0.2).

2. A test has 40 questions with 4 choices. Find the expected score from guessing.

10 Continuous variables and the normal distribution

The bell curve.

Many continuous measurements (heights, masses, errors, exam scores) are modelled by the normal distribution X ~ N(μ, σ2). Probabilities are areas under the curve, and P(X = a) = 0 for any single value.

Figure 1. About 68%, 95% and 99.7% of values lie within 1, 2 and 3 standard deviations of the mean.

11 Properties of the normal curve

Symmetric and bell-shaped.

The normal curve is symmetric about μ (mean = median = mode), bell-shaped, and its total area is 1. About 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.

Practice exercise 8

1. Heights are N(165, 72) cm. Estimate the percentage between 151 and 179 cm.

2. Estimate P(X > μ + σ).

12 Normal probabilities

Areas with the GDC.

WORKED EXAMPLE 8 Normal probabilities (standard)

Battery life is N(480, 352) hours. Find P(X < 450), P(X > 550) and P(460 < X < 500).

Solution

0.196A1

0.0228A1

0.432A1

Figure 2. P(460 < X < 500).

Common mistake

Entering the variance into the GDC. N(480, 352) has standard deviation 35.

Practice exercise 9

1. X ~ N(70, 102). Find P(X > 85).

2. Find the expected number of 200 batteries lasting under 450 hours.

13 Inverse normal calculations

From an area to a value.

WORKED EXAMPLE 9 Inverse normal (standard)

Find the battery life exceeded by 90% of batteries, and the top 5% threshold.

Solution

P(X < a) = 0.1: a = 435 hA1

P(X < b) = 0.95: b = 538 hA1

Practice exercise 10

1. Scores are N(62, 122). Find the score at the 80th percentile.

2. Find the interquartile range of N(50, 82).

14 Standardization and z-scores

How many standard deviations?

WORKED EXAMPLE 10 Comparing scores (standard)

Amir scored 74 in Maths (μ = 65, σ = 6) and 81 in English (μ = 75, σ = 8). In which did he do better relative to his group?

Solution

Maths z = 1.5; English z = 0.75A1 A1

MathsR1

Practice exercise 11

1. Find the z-score of 52 for N(60, 52).

2. A z-score of −1.2 in N(140, 152) corresponds to which value?

15 Finding an unknown mean or standard deviation

Inverse normal on Z.

WORKED EXAMPLE 11 Unknown σ (exam level)

Packets are N(500, σ2) g, and 3% weigh less than 490 g. Find σ.

Solution

z = invNorm(0.03) = −1.881A1

(490 − 500)/σ = −1.881M1

σ = 5.32 gA1

WORKED EXAMPLE 12 Unknown μ (exam level)

A machine fills bottles with σ = 4 ml. What mean is needed so that only 1% contain less than 750 ml?

Solution

z = −2.326A1

μ = 750 + 2.326(4) = 759.3 mlM1 A1

Practice exercise 12

1. X ~ N(μ, 92) and P(X > 100) = 0.2. Find μ.

16 Combining the normal and binomial distributions

One item, then many.

WORKED EXAMPLE 13 Normal then binomial (exam level)

Using batteries N(480, 352), a pack of 8 is bought. Find P(at least 2 last under 450 hours).

Solution

p = P(X < 450) = 0.1957A1

Y ~ B(8, 0.1957)M1

P(Y ≥ 2) = 0.484A1

Practice exercise 13

1. Eggs are N(60, 42) g; 'large' eggs exceed 63 g. Find P(exactly 4 large eggs in a box of 6).

17 Checking model assumptions

When is the model valid?

Before using a binomial model, check fixed n, constant p and independence. Before using a normal model, check that the data are roughly symmetric and bell-shaped (a histogram helps) and that impossible values (such as negative masses) have negligible probability.

Practice exercise 14

1. Explain why a normal model may not suit incomes.

2. A normal model for waiting times has μ = 3 min and σ = 2.5 min. Comment.

18 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 14 Mixed problem (exam level)

Commute times are N(35, 62) minutes. Find the probability that on 5 independent days the commute exceeds 40 minutes at most once.

Solution

p = P(T > 40) = 0.2023A1

Y ~ B(5, 0.2023): P(Y ≤ 1) = 0.733M1 A1

Practice exercise 15

1. X ~ N(100, 152). Find k with P(100 − k < X < 100 + k) = 0.9.

19 The Poisson distribution (HL)

Events in time or space.

The Poisson distribution models the number of events in a fixed interval when events occur randomly, independently and at a constant average rate m (calls per hour, typing errors per page). The mean scales with the interval.

WORKED EXAMPLE 15 Poisson (standard)

A help desk receives 4.5 calls per hour on average. Find P(exactly 3 calls in an hour) and P(more than 10 calls in 2 hours).

Solution

Po(4.5): 0.169A1

Po(9): 1 − P(X ≤ 10) = 0.294M1 A1

Practice exercise 16

1. Explain why the mean and variance being similar supports a Poisson model.

2. Typing errors occur at 0.8 per page. Find P(no errors on 3 pages).

20 Linear combinations of random variables (HL)

Sums and multiples.

A sum or difference of independent normal variables is also normal. Note that variances add even for a difference.

WORKED EXAMPLE 16 Sum of masses (exam level)

Apples are N(150, 122) g, the box N(200, 152) g. Find P(a box of 6 apples weighs more than 1150 g).

Solution

Total T: mean 6(150) + 200 = 1100A1

Var = 6(144) + 225 = 1089, SD 33M1 A1

P(T > 1150) = 0.0649A1

Practice exercise 17

1. X ~ N(30, 4), Y ~ N(25, 9) independent. Find P(X > Y).

21 Investigation: insurance premiums

A Paper 3 style exploration.

An insurer covers 10 000 cyclists. In a year, 4% make a claim; claims are $300 with probability 0.7 and $1500 with probability 0.3. 1. Find the expected claim cost per cyclist. 2. The insurer wants an expected profit of $20 per policy. Find the premium. 3. Model the number of claims with B(10 000, 0.04) and find its mean and SD. 4. Use a normal approximation to the number of claims to estimate the probability of more than 450 claims. 5. Discuss the insurer's risk.

Answers. 1. 0.04(0.7 × 300 + 0.3 × 1500) = 0.04 × 660 = $26.40. 2. $46.40. 3. mean 400, SD 19.6. 4. P(X > 450) ≈ 0.0054. 5. Claims are rarely far above the mean, but rare large events (correlated claims) are a risk.

22 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 1 style) (about 10 minutes)

X has distribution P(X = x) = 0.1, 0.2, 0.4, 0.3 for x = 0, 1, 2, 3.

(a) Find E(X). [2]

(b) A game pays $2X and costs $4. Find the expected gain and state whether the game is fair. [3]

Markscheme

Part	Answer	Marks
(a)	0 + 0.2 + 0.8 + 0.9	M1
	1.9	A1
(b)	E(2X) = 3.8	A1
	gain 3.8 − 4 = −0.2	A1
	not fair	R1

Examiner's comments. Expected gain must subtract the cost.

Question 2 (Paper 2 style) (about 15 minutes)

The masses of bags of rice are N(1002, 62) g.

(a) Find P(a bag weighs less than 990 g). [2]

(b) Find the mass exceeded by 85% of bags. [2]

(d) The mean is changed so that only 1% of bags weigh less than 990 g. Find the new mean. [3]

Markscheme

Part	Answer	Marks
(a)	normcdf	M1
	0.0228	A1
(b)	invNorm(0.15)	M1
	996 g	A1
(c)	B(20, 0.0228)	M1
	P(Y ≤ 1)	M1
	0.925	A1
(d)	z = −2.326	A1
	990 + 2.326 × 6	M1
	1004 g	A1

Examiner's comments. Keep p unrounded in (c).

Question 3 (Paper 2 style) (about 8 minutes)

A multiple-choice test has 25 questions with 5 options each. A student guesses.

(a) Find the expected number correct. [1]

(b) Find P(passing), where the pass mark is 10. [2]

Markscheme

Part	Answer	Marks
(a)	5	A1
(b)	1 − P(X ≤ 9)	M1
	0.0173	A1

Examiner's comments. Pass mark 10 means X ≥ 10.

23 Common misconceptions and links

Students often think…	Actually…
E(X) must be a possible value	it is a long-run average
P(X = a) > 0 for normal X	it is 0
Var(X − Y) = Var(X) − Var(Y)	variances add
any repeated experiment is binomial	check the conditions

How 4.7 connects to the rest of the course

Topic	Connection
4.5 Probability	events
4.8 Binomial distribution	details
4.11 Inferential statistics	hypothesis tests
4.2 Statistics	mean and SD

24 Check your understanding

Section A: multiple choice

1. ΣP(X = x) =

A. 1 B. 0 C. E(X) D. n

2. A fair game has expected gain

A. 1 B. the cost C. 0 D. positive

3. E(X) for B(30, 0.2) is

A. 4.8 B. 6 C. 30 D. 0.2

4. About what percentage lies within 2σ of μ?

A. 68% B. 99.7% C. 50% D. 95%

5. z = (x − μ)/σ measures

A. percentiles B. standard deviations from the mean C. variance D. probability

Section B: structured

6. X ~ N(40, 52). Find P(35 < X < 47). [2]

7. X ~ B(15, 0.3). Find P(X ≤ 3). [2]

Answers

1. A 2. C 3. B 4. D 5. B

6. 0.761

7. 0.297

25 Summary and topic map

Idea	Remember
Discrete	table; ΣP = 1
Expected value	ΣxP(X = x)
Fair game	E(gain) = 0
Binomial	B(n, p); np; np(1 − p)
Normal	N(μ, σ2); 68–95–99.7
Inverse normal	value from area
z-scores	(x − μ)/σ; unknown μ or σ

Before the exam, make sure you…

•define X and its distribution
•sketch and shade normal curves
•enter σ not σ2
•check model assumptions

Probability Distributions on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 3. Topic map for Probability Distributions.

26 Answers to practice exercises

Check your working against these answers. Full methods are expected in the exam.

Practice exercise 1

1. 0.2; 0.4

2. the probabilities sum to 1.2

Practice exercise 2

1. k(1 + 2 + 5) = 1: k = 1/8

Practice exercise 3

1. 3.5

2. 4.5

Practice exercise 4

1. $2.50

2. 1.75 − 5 = −$3.25

Practice exercise 5

1. A: 5600 − 600 = $5000: equal

Practice exercise 6

1. 0.215; 0.166

2. p is not constant (trials not independent)

Practice exercise 7

1. 2.4; 1.39

2. 10

Practice exercise 8

1. about 95%

2. about 16%

Practice exercise 9

1. 0.0668

2. 39.1

Practice exercise 10

1. 72.1

2. 10.8

Practice exercise 11

1. −1.6

2. 122

Practice exercise 12

1. 92.4

Practice exercise 13

1. p = 0.2266: 0.0237

Practice exercise 14

1. incomes are positively skewed

2. P(X < 0) = 0.115 is not negligible: unsuitable

Practice exercise 15

1. k = 24.7

Practice exercise 16

1. for Po(m), E = Var = m

2. e−2.4 = 0.0907

Practice exercise 17

1. X − Y ~ N(5, 13): 0.917

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