Combining events, and estimating from trials

What this lesson covers

In joint events, events can be dependent or independent.
Probabilities for joint events cannot simply be added, because doing so would double-count outcomes that are common to both events.
Mutually exclusive events cannot occur together.
The estimated probability of an event from an experiment is the number of times the event happens divided by the total number of trials in the experiment (i.e. the relative frequency for that event).

Key idea

Two questions answer all of this

Can both of these things happen at the same time?
And does one of them change the other?
Answer those in order and every question here follows.

Ask	Then
Can both happen at once?	no: you may add
	yes: adding double-counts
Does one affect the other?	no: multiply them
	yes: the second one has changed

Worked example

When you may simply add

Rolling a two, or rolling a five. Both cannot happen at once.
So the chance of either is a sixth plus a sixth. A third.
Events that cannot happen together are mutually exclusive.

Situation	Why	Answer
Rolling a 2, or a 5	cannot both happen
So add them	1/6 + 1/6	1/3
Heads or tails	cannot both happen
Mutually exclusive	is the name for that

Key idea

And when you may not

A card is drawn. What is the chance it is red, or a king?
Twenty-six red cards, four kings. So thirty out of fifty-two?
No. And the reason is worth counting rather than being told.

Worked example

Count the cards you counted twice

The king of hearts is red, and it is a king.
So is the king of diamonds. Both are in both lists.
Adding twenty-six and four counted those two cards twice.

Group	Working	Count
Red cards		26
Kings		4
Added	26 + 4	30
But red kings	counted twice	only 2 exist

Worked example

So take the overlap off once

Twenty-six plus four, minus the two counted twice.
Twenty-eight cards out of fifty-two, which is seven thirteenths.
Or count the actual cards. Same twenty-eight.

Step	Working	Result
Take the overlap off once	26 + 4 − 2	28
So P(red or king)	28 / 52	7/13
Or just count	the actual cards	28
Same answer	either way

Key idea

Which is why the rule exists

Adding is only safe when nothing is in both groups.
If something is, you have counted it twice and must take it off.
That is the whole of it. No formula needed.

Worked example

The second question: does one affect the other?

Two coins. The first one tells you nothing about the second.
So the chance of both heads is a half times a half. A quarter.
A six on a die and a head on a coin: a sixth times a half.

Events	Working	Answer
Two coins	the first tells you nothing
P(both heads)	1/2 × 1/2	1/4
A die and a coin	1/6 × 1/2	1/12
Independent	multiply them

Worked example

On a tree, you multiply along the path

Each branch carries its own probability.
Follow a path down and multiply what you pass.
A half, then a half, gives a quarter at the end of that path.

Worked example

When the first one does change things

A bag with five red and three blue. First draw: five out of eight.
Now keep it out. The bag has seven balls, four of them red.
So the second draw is four out of seven. Look in the bag again.

Step	The bag now	Chance
Bag: 5 red, 3 blue	P(red)	5/8
Keep it out	now 4 red of 7
P(red then red)	5/8 × 4/7	5/14
Put it back instead	5/8 × 5/8	25/64

Key idea

That is not a special kind of problem

A bag with a ball taken out is just a different bag.
You are not applying a dependent-events rule. You are looking again.
Put the ball back and the second draw is five eighths once more.

Key idea

Estimating from an experiment

How many times it happened, divided by how many trials.
That is the estimated probability, also called relative frequency.
Twenty-one sixes in a hundred rolls is nought point two one.

Idea	Detail
Estimated probability	times it happened ÷ trials
Also called	the relative frequency
21 sixes in 100 rolls	0.21
46 in 200 trials	0.23

Worked example

What a hundred rolls actually looks like

Expect about seventeen of each face. Here is a real run.
Fourteen, eighteen, fifteen, sixteen, sixteen, twenty-one.
Not one of them is seventeen, and not one is alarming.

Key idea

Compare every outcome, not just one

Looking only at sixes, twenty-one out of a hundred looks high.
Seen beside the other five, it is the top of a normal spread.
One outcome on its own cannot tell you whether a die is fair.

Summary

What to take away

Seven lines, and the first two are one question.


Ask: can both happen at once?	if not, you may add
If they can	adding counts the overlap twice
Mutually exclusive	means never together
Ask: does one affect the other?	if not, multiply
If it does	the second probability has changed
Estimated probability	is times happened over trials
More trials	a better estimate

Your turn

Your turn: four questions

Pause here and work all four.
Question 1 has an overlap in it.
Question 3 says the ball does not go back.

	Question
1	One card from 52. P(red or a king)?
2	Two coins. P(both heads)?
3	5 red, 3 blue. Two out, no replacing. P(red, red)?
4	46 times in 200 trials. Estimate?

Show answer

1. 28 out of 52, which is 7/13. There are 26 red cards and 4 kings, but two of the kings are red, so adding 26 and 4 counts those two twice. Take the overlap off once. 2. One quarter. The coins do not affect each other, so multiply a half by a half. 3. 5/14. The first draw is 5 out of 8. Then the bag has 7 balls and 4 of them are red, so the second is 4 out of 7. Multiply: 20 over 56, which cancels to 5 over 14. 4. 0.23, from 46 divided by 200. That is an estimate from this experiment, and a different 200 trials would give something a little different.

Worked example

The answers

Twenty-eight out of fifty-two, which is seven thirteenths.
A quarter.
Five eighths times four sevenths, which is five fourteenths.
Forty-six over two hundred, nought point two three.

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