Introduction to Probability

 Levels 5-7

 

The objectives of this unit are to:

*  find the probabilities of simple events using the idea of equally likely outcomes;

*  decide which of two events is most likely by comparing the sizes of their probabilities;

*  calculate the probability of an event not happening;

*  list all the outcomes when 2 or more events are combined;

*  begin to show events on sample space diagrams and use these to find probabilities.

 

Probability is the area of maths that examines how likely something (an event) is to happen.  At a basic level we can use words to describe how likely events are to happen:

            impossible- an event is impossible if it has no chance of happening;

            certain - an event is certain if it must happen;

            evens chance - an event  has an evens chance if it is as likely to happen as not to happen.


 

Examples:

*  It is impossible for a runner to run a marathon in less than one minute.

*  The probability of a ticket winning first prize in a raffle could be described as very unlikely.

*  If you threw a dice then it would be unlikely for it to land on a six.

*  Getting a red card from a pack of cards (with no jokers) would have an evens chance.

*  The probability of getting homework on a given school day might be described as likely.

*  Having something to eat tonight would be an event which would be very likely.

*  It is certain for Tuesday to follow Monday.

 


 

Probability scales

 

We can use numbers to give us a more precise way of measuring probability:

            *  An event which is impossible has probability 0.

            *  An event which is certain has probability 1.

            *  An evens chance corresponds to a probability of 1/2.

Probabilities can also be represented on a probability scale:

 

 

Example:  Mark the following 3 events on a probability scale:

a)  the probability that a baby is born a girl;

b)  the probability of throwing a six when a dice is thrown;

c)  the probability of drawing a card with a number on when picking a card from a pack of 52 cards.

 

Solution:

 

 


 

Skills recap: Simplifying fractions

 

A fraction can be simplified if there is a whole number (larger than 1) that divides into the numerator (top number) and the denominator (bottom number) exactly. 

 

Examples:

 

          

(divide top and bottom by 4)

          

(divide top and bottom by 2)

                

(divide top and bottom by 9)

          

(divide top and bottom by 7)

 


 

Equally likely outcomes

The probability of some events can be written down exactly if the outcomes are equally likely.

 

For example, if you throw a dice then there are 6 equally likely outcomes (it could land on a 1, 2, 3, 4, 5 or 6).  So the probability of getting a 6 is 1/6.

 

However, if a drawing pin is thrown the two outcomes (the point could land upwards or downwards) are not equally likely.  So it would NOT be true to say that the probability of the drawing pin landing point up is 1/2.

 

Example 1:

A box contains 5 yellow balls and 3 black balls.

If a ball is picked from the box without looking then it is equally likely to be any of the 8 balls.

So the probability of a yellow ball is 5/8.

The probability of picking a black ball is 3/8.

 

 

Example 2:

The following 10 cards are put into a hat.

 

10

 

11

 

11

 

12

 

13

 

15

 

16

 

17

 

20

 

24

 

If a card is picked out from the pack, it is equally likely to be any of the 10 cards. 

 

a)  The probability of picking out a card numbered 10 is 1/10.

b)  The probability of picking out a card with an odd number is 5/10 = 1/2.

c)  The probability of picking a card containing a multiple of 4 is 4/10 = 2/5 (the multiples of 4 are 12, 16, 20 and 24).

d)  The probability of picking out a card containing a number bigger than 12 is 6/10 = 3/5.

 

Example 3:

The spinner below is included in a board game.

 

 

If Jenny spins this spinner, the probability that is lands on ...

        blue is 2/6 = 1/3

        red is 3/6 = 1/2

        yellow is 0

        blue or red is 5/6.

 

Note:  Probabilities are usually expressed as fractions or decimals.  It is wrong to write a probability as a ratio (for example 1:3).

 


 

Probability of an event not happening.

 

If the probability of rain tomorrow is 3/10, then the probability that it won't rain must be 7/10.

 

In general,

        the probability of an event NOT happening = 1 - probability of the event happening.

 

Example:

A spinner is designed so that the probability of the pointer landing on red is 0.25.

So, the probability that the pointer does not land on red is 1 - 0.25 = 0.75.

 


 

Comparing the sizes of probabilities

 

Example

Julian likes orange sweets but does not like yellow sweets  He can pick a sweet from either bag A or bag B.

Bag A contains 3 orange sweets and 5 yellow sweets.

Bag B contains 4 orange sweets and 8 yellow sweets.

Which bag should he choose from in order that he has the biggest chance of getting an orange sweet.

 

Solution:

In bag A,  the probability of an orange sweet is given by: P(orange sweet) = 3/8

 

In bag B,  P(orange sweet) = 4/12 = 1/3

 

To decide which probability is the largest we should write these fractions with a common denominator so that they can be compared more easily.  Both denominators divide into 24 exactly, so 24  would be a suitable common denominator:

 

        3/8 = 9/24        (multiply top and bottom by 3)

 

        1/3 = 8/24        (multiply top and bottom by 8)

 

We can see that bag A gives the bigger probability of getting an orange sweet.

 


 

Listing outcomes

 

If an experiment is repeated 2 or more times we can sometimes list all the outcomes in a table.

 

For example, if a coin is thrown 3 times the outcomes would be:

1st coin

2nd coin

3rd coin

head

head

head

head

head

tail

head

tail

head

head

tail

tail

tail

head

head

tail

head

tail

tail

tail

head

tail

tail

tail

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

Example:

These two spinners are spun:

 

 

 

List the outcomes from the two spins.  Find the probability that you get:

a)  two reds

b)  the same colour from both spins.

 

Solution:

The outcomes from the two spins are listed below.

1st spinner

2nd spinner

red

red

red

white

red

blue

white

red

white

white

white

blue

 

 
 

 

 

 

 

 

 

 

 

 There are 6 equally likely outcomes.

 

a)  P(2 reds) = 1/6

b)  P(same colour) = 2/6 = 1/3          (2 reds or 2 whites)

 


 

Possibility space diagrams

 

Instead of listing the outcomes from two experiments, they can sometimes be displayed in a table.

 

For example, suppose that two dice are thrown.  The possible outcomes can be shown in the diagram below (called a possibility space diagram):

 

 

2nd dice

 

 

1

2

3

4

5

6

1st

dice

1

*

*

*

*

*

*

2

*

*

*

*

*

*

3

*

*

*

*

*

*

4

*

*

*

*

*

*

5

*

*

*

*

*

*

6

*

*

*

*

*

*

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

The table shows that there are 36 equally likely outcomes.

 


Example

A dice is numbered 1, 2, 2, 4, 5, 6.  It is thrown twice and the scores are ADDED.

Show the scores that can be obtained in a possibility space diagram.

Find the probability that the total score is a)  5  b) more than 8.

 

Solution:

The scores are shown below:

 

 

2nd throw

 

+

1

2

2

4

5

6

1st

throw

1

2

3

4

5

6

7

2

3

4

4

6

7

8

2

3

4

4

6

7

8

4

5

6

6

8

9

10

5

6

7

7

9

10

11

6

7

8

8

10

11

12

 

 
 

 

 

 

 

 

 

 

 

 

 


a)  P(total score is 5) = 2/36 = 1/18

b)  P(total score is more than 8) = 8/36 = 4/18 = 2/9.