Unit 26: Chance and Averages

Mean, median and mode

The mean is the total shared out equally. The median is the middle one when they are lined up in order. The mode is the one that appears most.

For 2, 3, 3, 4, 8: the mean is 20/5 = 4, the median is 3, and the mode is 3.

Three different answers to "what is typical", and all three are honest.

Why the median exists

Nine people earn £20,000. One earns £2,000,000.

The mean wage is about £218,000, which describes nobody in the room. The median is £20,000, which describes nine of them.

One extreme value drags the mean around and leaves the median untouched. That is the whole reason both exist.

Wanted, over total

When every outcome is equally likely, probability is counting:

probability = how many you want / how many there are

Rolling a 5 on a die: one outcome you want, six altogether, so 1/6.

Every probability sits between 0 (never) and 1 (certain).

The other thing that could happen

Everything either happens or it does not, so those two probabilities add to 1.

If rain has probability 0.3, then no rain has probability 0.7.

That is often the quick way in: counting what you *do not* want is sometimes far easier.

The mean when values repeat

Ten children own 0, 1, 2 or 3 pets. Rather than list all ten, a table gives each value and how often it happens.

To find the mean, multiply each value by its frequency, add those up, then divide by the total number of children.

So you are still doing "total ÷ how many" — the table just saves you writing the same number out four times.

The median in a table

With 10 values, the median sits between the 5th and 6th when they are in order.

Count along the frequencies: four children have 1 pet, so positions 1 to 4 are all 1s. The next three have 2, covering positions 5, 6 and 7.

So both the 5th and 6th are 2, and the median is 2. Counting along the table is much quicker than writing the list out.

A grid of two dice

Two dice give 36 outcomes: 6 for the first, 6 for the second.

Draw them as a 6 by 6 grid, with the total written in each square, and every probability question about two dice can be answered by counting squares.

The totals are not equally likely. There is one way to make 2, and six ways to make 7 — which is why 7 is the most common roll.

Multiply along, add across

A tree diagram puts a probability on each branch. Then:

Multiply along a path to get the chance of that whole sequence.
Add the paths that count as a win.

And every set of branches leaving one point must add to 1, which is a free check that you have not mislabelled anything.

With and without replacement

A bag has 3 red and 2 blue. Draw one, then another.

If you put the first back, the second draw is the same as the first: still 3 out of 5.

If you do not, the second branch changes — after a red, only 2 reds and 4 counters remain. Those changing numbers on the second layer are the whole difficulty of tree diagrams.

Probability times how many tries

If something has probability p and you try it n times, you expect it about p × n times.

Roll a die 60 times and expect about 10 sixes.

"About" is doing real work there. You will not get exactly 10, and if you did every time that would be far stranger than not.

Estimating a probability from results

Run it the other way. If a drawing pin landed point-up 36 times out of 60, estimate the probability as 36/60 = 0.6.

That is relative frequency, and it is how probability is measured for anything not perfectly symmetric — a bent coin, a real drawing pin, a football team.

More trials, better estimate. That is the only guarantee on offer.

What you win, weighted by how likely

A game costs 1 to play. You roll a die and win 6 if it shows a six, nothing otherwise.

One time in six you win 6, so on average you get back 6 × 1/6 = 1.

You pay 1 and get back 1. That is a fair game — over many plays, nobody gains. Multiply each prize by its probability and add: that total is what to expect.

Adding up several prizes

A spinner pays 10 with probability 1/10, 2 with probability 1/2, and nothing the rest of the time.

Average return: 10 × 1/10 + 2 × 1/2 = 1 + 1 = 2.

So it is worth playing if it costs less than 2, and not if it costs more. That single number decides it — and it is how insurance and casinos are both priced.

And means multiply

For two things that do not affect each other, the chance of both happening is the two probabilities multiplied.

Two heads in a row: 1/2 × 1/2 = 1/4.

It is the multiplication principle from unit 25 again — count all the outcomes, count the ones you want.

Or means add — carefully

For things that cannot both happen, the chance of one or the other is the two added.

Rolling a 1 or a 2: 1/6 + 1/6 = 1/3.

But if they *can* both happen you would count the overlap twice, so you have to take it off again. Rolling an even number or a number over 3: 3/6 + 3/6 - 2/6 = 4/6.