Unit 21 Section 7 : General Addition Law
Events may not always be mutually exclusive.
Example 1
A normal pack of cards contains 52 cards. Some are red, some are picture cards, and some are red picture cards.
This means that a card picked at random could be both a red card and a picture card.
The events 'red card' and 'picture card' are therefore not mutually exclusive.
General Addition Law for Events which are not Mutually Exclusive
If there are two events, A and B, which are not mutually exclusive, then:
P(A or B) = P(A) + P(B) – P(A and B)

Example 2
One of the numbers from 1 to 10 is selected at random. We want to find P(even OR greater than 6).
A venn diagram can be useful in this situation:
We can now see that if we simply added P(even) and P(greater than 6) we would be including 8 and 10 twice.
Using the General Addition Law above, we can see that:
P(even OR greater than 6) = P(even) + P(greater than 6) – P(even AND greater than 6)
We can therefore work out P(even OR greater than 6) in stages:
P(even) = 
5  
10 
P(greater than 6) = 
4  
10 
P(even AND greater than 6) = 
2  
10 
P(even OR greater than 6) = P(even) + P(greater than 6) – P(even AND greater than 6) = 
5 
+ 
4 
– 
2 
= 
7     
10 
10 
10 
10 
Practice Question
Work out the answers to this question then click on the buttons marked
to see whether you are correct.
A normal pack of 52 playing cards has four 'suits' which are clubs, diamonds, hearts and spades.
Clubs and spades are black cards, and diamonds and hearts are red cards.
Each 'suit' contains 13 cards: an Ace, the numbers 2 to 10, and three picture cards which are Jack, Queen and King.
A card is picked at random from the pack and we want to find P(red OR picture card)
(a) What is P(red)?
(b) What is P(picture card)?
(c) What is P(red AND picture card)?
(d) What is P(red OR picture card)?
NOTE: We could have worked out the above answer by counting up the cards which are red cards or picture cards
(or both) and then put the total as a fraction over 52, but the example was to illustrate the General Addition Law
for events which are not mutually exclusive. It is not always possible to work the answer out by counting.
Exercises
Work out the answers to the questions below and fill in the boxes. Click on the
button to find out whether you have answered correctly. If you are right
then will appear and you should move on to the next
question. If appears then your answer is wrong. Click
on to clear your original answer and have another go.
If you can't work out the right answer then click on
to see
the answer.
Make sure you simplify all the fractions in your answers.
You have now completed Unit 21 Section 7
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