GCSE MATHSGCSE Session 27 – Probability and Relative Frequency

Frequency Polygons This is basically a line graph plotted from the mid

points of each bar of the histogram.

The reason they are used is that more than one polygon can be plotted on the same set of axis, and the distributions can be compared

Matching histogram and frequency polygon shown on page 422

Ex 38.3 Q1 and Q5 (for Q5 consider where the bars would be to help you visualise)

Foundation learners complete the rest of Ex 38.8

Histograms (unequal) Higher For unequal class widths, we can no longer

show frequency on the y axis. Instead this is replaced with frequency density (frequency per unit of the class width)

Frequency is proportional to the AREA of the bar, not the height

The modal class is the highest bar. To find the mode from a frequency table of un equal class widths, you would need to work out frequency density for each class then select the one with the highest FD.

Calculating frequency density Frequency = Frequency Density x Class Width

Frequency ÷ Class Width = Frequency Density

Remember one of these and be able to rearrange it into the other. When you get the hang of histograms these will be 2nd nature

You will need to extend the frequency table with unequal class widths to include the class width interval and then work out the frequency density in order to plot a histogram. Example p425

Ex 38.4 – also read the what you need to know.

Probability

The chance that an event will occur

The different possible things which might happen are called outcomes

Described in words or on a scale

Impossible, unlikely, even, likely, certain

The scale runs from 0 to 1 and can be represented in fractions or decimals. Also percentage can be used (0%-100%)

So a probability will be represented as a fraction or decimal less than or equal to 1 or percentage less than 100%.

If you do a probability calculation and it comes out as 1.4 for example, you must have made a mistake

Discussion of Ex40.1

Equally likely outcomes E.g heads and tails are equally likely. 12345 or 6

and equally likely when rolling one dice.

Random or fair means the probabilities are equally likely.

P(x) means the probability of x happening P(an odd number) is the probability of rolling an

add number on a dice

Probability (x)=

Ex40.2 Q4 and Q7 and Q10 – 5 min

Relative Frequency

Relative Frequency = Frequency = number of times an even occurs Relative = relative to the number of trials

We may do experiments, look at surveys or use past records to generate RF.

Relative Frequency is used to estimate probability.

We do not use P(x) = when referring to relative frequency, because it is not an exact calculation.

As the number of trials increase, RF gives a better estimate of probability

In the example (p445) Jamie calculates the probability every 10 trials, using ALL the previous data, so his trial number keeps increasing and therefore his RF gets closer to the calculated probability.

In most cases we use RF because we cannot calculate the probability, this experiment is just to illustrate the point that as the number of trials increase, RF gives a better estimate of probability

Ex 40.3 – 10 minutes

Mutually exclusive events Events that cannot happen at the same time

are called mutually exclusive. The occurrence of one means the others cannot have occurred.

If you roll a 3 on a dice, you cannot have rolled a 1245 or 6 in the same trial.

Probability of events not happening

Any given event can be said to occur, or not occur.

The probability of an even not occurring is 1 – the probability of it occurring.

P (not A) = 1 – P(A)

Technically known as the compliment of A. It includes every other event which is not A, this is why they add up to 1 (certainty).

OR is a case where either is required so both probabilities are added together.

Ex 40.4 – 10 minutes

AND is a case where both are required, so the probabilities are multiplied . (You cannot have an AND situation with mutually exclusive events) These events must be independent.

For independent events, it’s as simple as just multiplying the 2 probabilities. With dependent events you will need to construct a tree diagram which we will look at later.

Discuss example on page 449 Ex 40.5 - 10 minutes (Foundation learners continue with this and previous

examples)

Independent and dependent (conditional) events

Independent events are when the probability of an event is unaffected by the previous or other events.

Conditional events are where one event has an affect on the probability of another e.g. when items are not replaced, as in a lottery or card game. (we will look more at this in the final exercise)

Tree Diagrams - Higher

A ball is picked from the bag then replaced, and a second pick is taken.

Fill in the tree diagram

What’s the probability of…?P(RR) =

P(RB or BR) =

EX 40.6 – 5 minutes

Example of conditional probability (on next slide)

Ex 40.7 – rest of session (You can go back and do more Ex 40.6 if

you choose,

Tree Diagrams (2) - Higher

A ball is picked from the bag but NOT REPLACED, and a second pick is taken.

Fill in the tree diagram

What’s the probability of…?P(RR) =

P(RB or BR) =