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 Click on this button below to see the correct answer 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 Click this button to see if you are correct 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 Click on this button to see the correct answer to see the answer.

Make sure you simplify all the fractions in your answers.

Question 1
One of the numbers 10 to 20 is selected at random.

Work out the probability that it is:
(a) an even number
(b) a multiple of 5
(c) an even number AND a multiple of 5
(d) an even number OR a multiple of 5

Question 2
The numbers 1 to 10 are sorted into sets as shown in the Venn diagram:

One of these numbers is selected at random.

Work out the probability that it is in:
(a) A
(b) B
(c) A and B
(d) A or B

Question 3
A fair dice is rolled.

What is the probability of getting a prime number or an odd number?

Question 4
One of the numbers 1 to 20 is selected at random.

What is the probability that the number is a multiple of 3 or a multiple of 4?

Question 5
A bingo set contains 100 balls each marked with one of the numbers 1 to 100.
One of these balls is selected at random.

What it the probability that the number on this ball is a multiple of 7 or a multiple of 10?

Question 6
A and B are two possible outcomes in an experiment.

If P(A) =  3 , P(B) =  3 , and P(A and B) =  1 , find P(A or B)
4 8 3

Question 7
A and B are two possible outcomes in an experiment.

If P(A or B) =  3 , P(A and B) =  2 , and P(A) =  1 , find P(B)
5 5 2

A and B are two possible outcomes in an experiment.

If P(A or B) =  1 , P(A) =  2 , and P(B) =  1 , find P(A and B)
2 5 3

Question 9
The probability that Jai is late for school is 0.1.
The probability that he forgets his lunch is 0.3.
The probability that he forgets his lunch and is late is 0.05.

What is the probability that he forgets his lunch or is late? (give your answer as a decimal)

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