Course

Cambridge A Level Mathematics: Probability & Statistics 1 (9709 Paper 5)

Name: Cambridge A Level Mathematics: Probability & Statistics 1 (9709 Paper 5)
Price: 2199 BDT

Paper 5 of 9709 from the ground up: data diagrams, averages and spread, probability, permutations and combinations, discrete, binomial and geometric distributions, and the normal distribution.

100 lessons18h 13mEnglish

What you'll learn

Draw and read stem-and-leaf diagrams, box-and-whisker plots, histograms and cumulative frequency graphs
Find the mean, median, mode, interquartile range and standard deviation from data, tables and given totals
Work out probabilities with sample spaces, Venn diagrams, tree diagrams and conditional probability
Count arrangements and selections with permutations and combinations, including restrictions
Use distribution tables, the binomial and geometric distributions and their expectations
Find normal probabilities and values with the table, and approximate a binomial with a continuity correction

Requirements

IGCSE or O Level Mathematics, or equivalent
Comfortable with fractions, decimals, percentages and reading a graph
Able to solve a pair of simultaneous linear equations

Course content

100 lessons · 18h 13m

5.1 · Representation of data

Sort data into types, then draw it: stem-and-leaf diagrams, histograms with frequency density, and cumulative frequency graphs, and choose the diagram that suits the question.

0/12

Types of data

Data is qualitative or quantitative, and quantitative data is discrete (counted) or continuous (measured).

Preview10m 2.

Stem-and-leaf diagrams

A stem-and-leaf diagram shows the shape of the data like a bar chart but keeps every value, read with a key.

Preview12m

Back-to-back stem-and-leaf diagrams

Two sets of data share one stem, leaves on each side, so the two groups can be compared at a glance.

12m

Grouped data and class boundaries

Class boundaries, class widths and mid-values turn a long list of measurements into a grouped frequency table.

12m

Histograms

When classes have different widths, the bar height is frequency density, so the area of each bar is its frequency.

13m

Interpreting histograms

Read each frequency back as an area, and estimate how many values lie in part of a class.

12m

Cumulative frequency graphs

Keep a running total of the frequencies and plot each total at the upper class boundary.

12m

Using a cumulative frequency graph

Read how many values lie below, above or between two values, and turn that count into a proportion.

12m

Choosing a suitable representation

Every diagram keeps some information and loses some. Choose the one that answers the question being asked.

12m

10.

Representation of data: practice

A timed drill of mixed questions on types of data, stem-and-leaf diagrams, histograms and cumulative frequency.

12m

11.

Representation of data: exam practice

Paper 5 style questions with full worked solutions and the marks each step earns.

15m

12.

Representation of data: summary

Every diagram in the chapter, the frequency density rule and the common mistakes on one screen.

5.1 · Measures of central tendency

Find the mode, the mean and the median of raw, tabled and grouped data, work the mean out from given totals and coded totals, and choose the average that suits the data.

0/12

The mode and the modal class

The mode is the value that appears most often. For grouped data the modal class has the highest frequency density.

Preview10m 2.

The mean

Add every value and divide by how many there are. The mean is the point where the data balances.

Preview12m

The mean of a frequency table

Each value counts as many times as its frequency, so the mean is the sum of fx divided by the sum of f.

12m

The mean of grouped data

The exact values are lost in a grouped table, so each class is represented by its mid-value and the mean is an estimate.

12m

The mean from given totals

Given the sum of x and n, the mean follows at once, and the totals of two groups add to give the mean of both.

12m

The mean from coded totals

Subtract a number a from every value, find the mean of what is left, then add a back.

12m

The median

Put the values in order: the median is the middle one, or halfway between the two middle ones.

12m

The median of grouped data

Go across at half the total on a cumulative frequency graph and read down, or work it out inside the median class.

12m

Choosing a measure of central tendency

One extreme value drags the mean and barely moves the median, so the shape of the data decides which average to use.

12m

10.

Measures of central tendency: practice

A timed drill of mixed questions on the mode, the mean from lists, tables and totals, and the median.

12m

11.

Measures of central tendency: exam practice

Paper 5 style questions on averages, with full worked solutions and the marks each step earns.

15m

12.

Measures of central tendency: summary

The three averages, every mean formula and the common mistakes on one screen.

5.1 · Measures of variation

Measure how spread out data is: the range, the interquartile range and box-and-whisker plots, outliers, and the variance and standard deviation from data, grouped tables, totals and coded totals.

0/15

The range

Two sets of marks can share a mean and still differ. The range, largest minus smallest, is the first measure of spread.

Preview10m 2.

Quartiles and the interquartile range

The quartiles cut the ordered data into four equal parts, and the interquartile range holds the middle half.

Preview12m

Box-and-whisker plots

Five numbers draw the whole plot: the smallest value, the lower quartile, the median, the upper quartile and the largest value.

12m

Outliers

A value more than 1.5 times the interquartile range beyond a quartile is an outlier, and the plot marks it on its own.

12m

Quartiles and percentiles from a cumulative frequency graph

Go across at a quarter, three quarters or any percentage of the total, read down, and the width between gives the spread.

12m

Variance and standard deviation

Square each distance from the mean, average the squares, take the square root: a spread that uses every value.

13m

Variance using the sum of the squares

The mean of the squares minus the square of the mean gives the same variance with much less working.

12m

Standard deviation of grouped data

Use the mid-value of each class, counted f times, to estimate the standard deviation of a grouped table.

13m

Mean and standard deviation from totals

Given n, the sum of x and the sum of x squared, you can find the mean and the standard deviation without the data.

12m

10.

Mean and standard deviation from coded totals

Subtracting a from every value moves the mean by a and leaves the standard deviation unchanged.

13m

11.

Combining two sets of data

Add the two values of n, the two sums of x and the two sums of x squared, then find the mean and spread of everyone.

13m

12.

Comparing two sets of data

Compare one average and one measure of spread, and say what each difference means in the context.

12m

13.

Measures of variation: practice

A timed drill of mixed questions on the interquartile range, outliers, box plots and standard deviation.

12m

14.

Measures of variation: exam practice

Paper 5 style questions on spread, including coded totals and two data sets, with full worked solutions.

15m

15.

Measures of variation: summary

Every measure of spread, every standard deviation formula and the common mistakes on one screen.

5.3 · Probability

Count equally likely outcomes, then combine events: the complement, mutually exclusive and independent events, tree diagrams, drawing with and without replacement, and conditional probability.

0/15

Experiments, outcomes and events

List every outcome of an experiment, and when all are equally likely, probability is a count divided by a count.

Preview10m 2.

Sample space diagrams

A grid of every result for two dice turns a question like "total score 8" into counting cells.

Preview12m

Equally likely outcomes: drawing from a bag

Every ball in the bag is equally likely, so the probability of a colour is its count over the total.

12m

The complement of an event

A' is everything that is not A, so P(A') = 1 − P(A), and 'at least one' is often quickest this way.

11m

Mutually exclusive events

Events that cannot happen together have no overlap, so P(A or B) is P(A) + P(B).

12m

Events that are not mutually exclusive

When two events overlap, adding counts the overlap twice. Count it once from a Venn diagram or a grid.

12m

Independent events

If one event does not change the chance of the other, P(A and B) is P(A) × P(B).

12m

Testing whether two events are independent

Work out P(A ∩ B) and P(A) × P(B) separately. Equal means independent, different means not.

12m

Tree diagrams

Multiply along the branches for "and", add the ends of branches for "or".

12m

10.

With and without replacement

Putting the ball back keeps the draws independent. Keeping it out changes the second set of branches.

12m

11.

Conditional probability

Given that B has happened, count only inside B: P(A | B) = P(A ∩ B) ÷ P(B).

12m

12.

Conditional probability from a tree diagram

Told the result of the second stage, find the chance of the first: divide one branch end by the total of the matching ends.

13m

13.

Probability: practice

A timed drill of mixed questions: grids, bags, Venn diagrams, independence, trees and conditional probability.

12m

14.

Probability: exam practice

Paper 5 style probability questions with full worked solutions and the marks each step earns.

15m

15.

Probability: summary

Every probability rule, the independence test and the common mistakes on one screen.

5.2 · Permutations and combinations

Count without listing: arrangements in a line with repeats and restrictions, permutations and combinations, selections with conditions, and probabilities worked out by counting.

0/12

Arrangements in a line and factorial notation

Fill the first place n ways, the next n − 1 ways, and so on. The product is n factorial, written n!.

Preview12m 2.

Arrangements with repeated objects

Swapping two identical letters makes no new word, so divide n! by the factorial of each repeat.

Preview12m

Arrangements where objects must be together

Tie the objects into one block, arrange the blocks, then arrange the objects inside the block.

12m

Arrangements where objects must not be together

Arrange the others first and put the separated objects in the gaps, or subtract the "together" count from the total.

12m

Arrangements with restrictions on position

Fill the restricted places first, such as the two ends of a line or a front row, then arrange everyone else.

12m

Permutations

Arranging r objects chosen from n gives nPr = n! ÷ (n − r)! orders, because the order matters.

12m

Combinations

When order does not matter, each group of r was counted r! times, so nCr = nPr ÷ r!.

12m

Selections with restrictions

Split the selection into cases, multiply the choices within each case and add the cases.

13m

Probability using permutations and combinations

Count the favourable arrangements or selections, divide by the total, and the probability follows.

13m

10.

Permutations and combinations: practice

A timed drill of mixed questions: arrange or select, repeats, together, apart and fixed places.

12m

11.

Permutations and combinations: exam practice

Paper 5 style counting questions with full worked solutions and the marks each step earns.

15m

12.

Permutations and combinations: summary

n!, nPr and nCr, the method for each restriction and the common mistakes on one screen.

5.4 · Probability distributions

Turn the outcomes of a game into a discrete random variable, draw up its probability distribution table, and calculate its expectation and variance.

0/8

Discrete random variables

A random variable gives each outcome a number, such as the total on two dice or the prize won.

Preview11m 2.

Probability distribution tables

List every value of X with its probability. The probabilities in the table always add up to 1.

Preview12m

Probability distributions with an unknown

When one probability is written with a letter, use the total of 1 to find it.

12m

Expectation

E(X) is the sum of each value times its probability: the mean result over a very long run of games.

12m

Variance of a discrete random variable

Var(X) = E(X²) − [E(X)]², the same idea as the variance of data with probabilities in place of frequencies.

13m

Probability distributions: practice

A timed drill: build the table, find the unknown, then E(X) and Var(X).

Probability distributions: exam practice

Paper 5 style questions on distribution tables, E(X) and Var(X), with full worked solutions.

15m

Probability distributions: summary

The table, the total of 1, E(X), Var(X) and the common mistakes on one screen.

5.4 · The binomial and geometric distributions

Recognise when a count of successes is binomial and when a wait for the first success is geometric, calculate their probabilities, and use their expectation and the binomial variance.

0/11

The binomial distribution

A fixed number of independent trials, each a success or a failure with the same chance, gives X ~ B(n, p).

Preview12m 2.

Calculating binomial probabilities

P(X = r) = nCr × p^r × (1 − p)^(n − r): nCr counts the orders, the powers give the chance of one order.

Preview13m

Binomial probabilities: at most and at least

Add the single probabilities for "at most", and use 1 minus the rest for "at least".

12m

Expectation and variance of the binomial distribution

For X ~ B(n, p), E(X) = np and Var(X) = np(1 − p), with no table to add up.

12m

The geometric distribution

Count the trials up to and including the first success: P(X = r) = (1 − p)^(r − 1) × p.

12m

Geometric probabilities over a range of values

More than r trials means the first r all failed, so P(X > r) = (1 − p)^r.

12m

Expectation of the geometric distribution

For X ~ Geo(p), E(X) = 1/p: with a 1 in 6 chance, the first six comes on the sixth throw on average.

11m

Choosing a binomial or geometric model

A fixed number of trials is binomial, waiting for the first success is geometric, and some situations are neither.

12m

The binomial and geometric distributions: practice

A timed drill: pick the model, then exactly, at most, at least and the expectation.

12m

10.

The binomial and geometric distributions: exam practice

Paper 5 style questions on B(n, p) and Geo(p), with full worked solutions and mark schemes.

15m

11.

The binomial and geometric distributions: summary

The conditions, the two formulas, np, np(1 − p), 1/p and the common mistakes on one screen.

5.5 · The normal distribution

Model a continuous variable with a normal curve, find probabilities and values with the table, find an unknown mean or standard deviation, and approximate a binomial with a continuity correction.

0/7

The normal distribution

Many measurements pile up in a bell shape around the mean. For a continuous variable, probability is area under the curve.

Preview12m 2.

The standard normal distribution and tables

Z ~ N(0, 1) has one table: Φ(z) gives the area to the left of z.

Preview12m

Using symmetry: negative z and areas between values

The table prints only positive z. Symmetry gives Φ(−z) = 1 − Φ(z), and subtraction gives the area between.

12m

Standardising a normal variable

Z = (X − μ) ÷ σ counts how many standard deviations a value is from the mean.

12m

Finding probabilities for any normal distribution

Sketch the curve, standardise, read the table and use symmetry: P(X > x₁) for X ~ N(μ, σ²).

13m

Finding a value from a probability

Read the table backwards to get z, then x = μ + zσ.

12m

Finding an unknown mean or standard deviation

Given one probability, the z-value gives one equation, enough to find a missing μ or σ.

12m

About the creator

8bitlesson

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