GCSE Maths

Mutually Non-Exclusive Events - Events that CAN happen at the same time

For example, pretend the blue represents rain, and the yellow represents sun. Sometimes it can rain and be sunny at the same time. That is when the two come together in the middle.

What if the chance of rain is 30% and the chance of sun is 80%. What is the chance of both rain and sun?

The probability of rain is 0.3

The probability of sun is 0.8

NORMALLY you multiply probabilities when finding the possibility of both occurring at the same time:

0.3 x 0.8  =  0.24

BUT since there is an overlap, where rain and sun can occur together, this is not a simple matter of multiplying the probabilities together.

Look at the diagram with the circles. What can be said? Everything must add up to 1. So how do you add up the areas?

Let’s say the blue circle for rain is P(A)

And the yellow circle for sun is P(B)

And the overlap of both rain and sun is P(AnB)      – the ‘n’ is like saying ‘and’

First, you add the two circles together.

P(A) + P(B)

Then, because you have counted the overlap twice, you have to take it away once.

P(A) + P(B) – P(AnB)

All of this must equal 1, so:

P(A) + P(B) – P(AnB) = 1

Now, to rearrange this, to find what the overlap is:

P(AnB) = P(A) + P(B) – 1

.

Going back the the question of rain and sun:

P(A) = 0.3          -   rain

P(B) = 0.8           -   sun

.

So for rain and sun, we use the equation we found:

P(AnB) = 0.3 + 0.8 – 1  =  1.1 – 1  =  0.1

So the probability of both rain and sun at the same time is 10%

 

Mutually Exclusive Events – Events that CANNOT happen at the same time

For example, pretend the blue represents rain drops and the yellow represents COMPLETELY dry weather. It is either wet or dry, but it cannot be both wet and dry at the same time. The events are EXCLUSIVE. So there is no overlap. The chances of A and B are zero:

P(AnB) = 0       

The equation we found thus becomes

P(A) + P(B) = 1

The chance of dry weather is

P(B) = 1 – P(A)

If the chances of dry weather tomorrow is 99%,  what is the chance of it being wet? Of course almost ZERO. That might be a desert.

So If the chance of it being COMPLETELY DRY in England tomorrow is 30% what is the chance of rain drops? Quite clearly the OTHER PART of that, which is 70%. How did we figure that out? Almost intuitively, just as clearly as we know that two halves make a whole.

30% is 0.3

So to find the other part:

P(B) = 1 – P(A) = 1 – 0.3  =  0.7

Which is 70%. The chances of rain tomorrow is 70%.

And of course, the chances of both rain and no rain is 0%  (unless they happen one after the other)

 

SUMMARY – THE PROBABILITY OF TWO EVENTS HAPPENING TOGETHER

Before you assume that you have to multiply the probabilities, always ask – Is there an overlap?

If there is and overlap, and they can happen together, you NEVER multiply the probabilities, but you ALWAYS solve for P(AnB) in       

P(A) + P(B) – P(AnB) = 1

If there is no overlap, and the circles do not touch, then there is obviously no possibility that the events can happen together. But, they can happen one after the other. You MULTIPLY the two probabilities to find the probability of both. In the example above, clearly there is no possibility of both wet and dry weather at the same time. We therefore have to interpret it to mean that it is dry AND THEN wet. The chance of this is 

0.3 x 0.7 = 0.21   or 21%.

 

THE FOUR POSSIBILITIES IN THE WEATHER

Let us recall our weather example again:

It is easy to see that this represents a sky with the rain without sun (dark blue) or sun without rain (yellow) or both at the same time (the green in the middle)

Or no sun, and no rain, but just clouldy (the light blue surrounding the circles), marked X. When we looked at this the first time, we assumed this was zero. But we have to admit this possibility.

We have now added one more possible outcome. There are now 4 bits of information shown.

Let us now list THE POSSIBLE OUTCOMES – Just Count All the colors:

 

THE DARK BLUE – Possibility of Rain without sun = P(A) – P(AnB)     

THE YELLOW – Possibility of Sun without rain = P(B) – P(AnB)

THE GREEN – Possibility of rain and sun = P(AnB)

THE LIGHT BLUE – Possibility of no rain and no sun = P(X)

 

When we put all four together, we can write it like this:

P(A) – P(AnB)       +       P(B) – P(AnB)       +       P(AnB)         +        P(X)       =   1 

They all equal 1 because they are the only possible outcomes. Simplifying:

P(A) + P(B) – P(AnB) + P(X) = 1

 

QUESTION: The possibility of rain is 30%, the possibility of sun is 80 %, The possibility of rain and sun together is 15%. What is the possibility that there will be no rain, and no sun (just cloudy sky)?

P(A) = 30%      rain

P(B) = 80%      sun

P(AnB)  =    15%

P(X) = ?              no sun and no rain at all

ANSWER - Rearranging the equation:

P(X) = 1 – P(A) – P(B) + P(AnB) = 1 – 0.3 – 0.8 + 0.15 = 0.05

Therefore, there is 5% chance of no sun and no rain at all.

To solve this equation, you need to know 3 out of 4 bits of information.

 

SIMPLICITY

GCSE Maths just covers events where all the parts add to the whole separately, and there is no possibility of ‘rain and sun’ together. They are called ‘Mutually Exclusive’:

 

NON-REPLACEMENT – INDEPENDENT AND DEPENDENT

In the case of taking a ball out of a bag, the probability depends on the total number of balls. If the ball is placed back again, the probability does not change the next time a ball is taken out. The probabilities of taking a ball out twice, in this way, replacing it each time, are INDEPENDENT.

In the case of taking a ball out and not replacing it, clearly the number of balls is reduced by 1, so the probability changes. The probabilities of taking a ball out twice, in this way, NOT replacing it each time, are DEPENDENT.

It is called ‘dependent’ because the probability is dependent on the event happening before.