Independent Events

Probability

unlikely50-50

chance likely

possible

probable

certain

poor chance

A sample space is the set of all possible outcomes.A sample space is the set of all possible outcomes.

Vocabulary

A simple example is flipping a coin.

The sample space is {heads, tails}

Vocabulary

Examples are rolling two dice, or spinning a spinner and rolling a dice

Two events are independent if the outcome of one has no effect on the outcome of the other.Two events are independent if the outcome of one has no effect on the outcome of the other.

Vocabulary

Examples are picking two marbles out of a bag, or two socks out of a drawer

Two events are dependent if the outcome of one event relies on the other event.Two events are dependent if the outcome of one event relies on the other event.

Vocabulary

The complement of an event is all the outcomes NOT included in the event. It is shown by A’.The complement of an event is all the outcomes NOT included in the event. It is shown by A’.

What is the probability of it not landing on the yellow sector?

P(not yellow) OR P(yellow’) =34

The following spinner is spun once:

Vocabulary

The intersection of two events is all the outcomes that are SHARED by both events. It is denoted by A∩B and can be read A and B

The intersection of two events is all the outcomes that are SHARED by both events. It is denoted by A∩B and can be read A and B

A ∩B is the number 6P(A ∩B) = 1/6

Event A is even numbers on a diceEvent B is multiples of 3 on a dice

If two events are independent, then their intersection can be calculated as P(A ∩B)=P(A)*P(B)If two events are independent, then their intersection can be calculated as P(A ∩B)=P(A)*P(B)

Vocabulary

The union of two events is all the outcomes of either event events. It is denoted by AUB and can be read A or B.The union of two events is all the outcomes of either event events. It is denoted by AUB and can be read A or B.

A UB is {1,3,5,6}P(A UB) = 4/6 or 2/3

Event A is odd numbers on a diceEvent B is multiples of 3 on a dice

Union can be calculated as P(AUB)=P(A)+P(B)- P(A∩B)Union can be calculated as P(AUB)=P(A)+P(B)- P(A∩B)

© Boardworks Ltd 2005 8 of 55

BAB or A

This symbol means “union”

Consider a marriage or union of two people –

when two people marry, what do

they do with their possessions ?

The bride takes all her stuff & the groom takes all his stuff & they put it together!

And live happily ever after!This is similar to the union

of A and B.All of A and all of B are put

together!

Example

• The two spinners are spun. What is the probability that both spinners will show an even number?

P(1st spinner even) = 4 8

P(2nd spinner even) = 48

P(both spinners even) =4 ∙ 4 = 1 8 8 4

Example

• A game uses a dice and a spinner.

1. A player rolls a dice. What is the P(odd #)?

2. The player spins the spinner. What is the P(red)?

3. What is the P(odd # and Red)?

Example• There are 4 red, 8 yellow and 6 blue socks in a

drawer. Once a sock is selected it is not replaced. Find the probability that 2 blue socks are chosen.

P(1st blue sock) = 6 18

P(2nd blue sock) = 517

P(Two blue socks) = 6 ∙ 5 = 5 18 17 51

# of socks after 1 blue is removed

Total # of socks after 1 blue is removed

Example

• In a certain town, the probability that a person plays sports is 65%. The probability that a person is between the ages of 12 and 18 is 40%. The probability that a person plays sports and is between the ages of 12 and 18 is 25%. Are the events independent?

Mutually exclusive outcomes

Outcomes are mutually exclusive if they cannot happen at the same time.

For example, when you toss a single coin either it will land on heads or it will land on tails. There are two mutually exclusive outcomes.

Outcome A: Head

Outcome B: Tail

Mutually exclusive outcomes

A pupil is chosen at random from the class. Which of the following pairs of outcomes are mutually exclusive?

Outcome A: the pupil has brown eyes.Outcome B: the pupil has blue eyes.

Outcome C: the pupil has black hair.

Outcome D: the pupil has wears glasses.

These outcomes are mutually exclusive because a pupil can either have brown eyes, blue eyes or another colour of eyes.

These outcomes are not mutually exclusive because a pupil could have both black hair and wear glasses.

Adding mutually exclusive outcomes

If two outcomes are mutually exclusive then their probabilities can be added together to find their combined probability.

What is the probability that a card is a moon or a sun?

P(moon) =13

and P(sun) =13

Drawing a moon and drawing a sun are mutually exclusive outcomes so,P(moon or sun) = P(moon) + P(sun) =

13

+13

= 23

For example, a game is played with the following cards:

Adding mutually exclusive outcomes

If two outcomes are mutually exclusive then their probabilities can be added together to find their combined probability.

For example, a game is played with the following cards:

What is the probability that a card is yellow or a star?

P(yellow card) =13

and P(star) =13

Drawing a yellow card and drawing a star are not mutually exclusive outcomes because a card could be yellow and a star.

P (Y U S) = P(Y) + P(S) – P(Y∩S) = 1/3 + 1/3 – 1/9 = 5/9.

Venn Diagrams

• Venn diagrams are useful in figuring out probabilities

P(AUB)’

Example

• If A is the students who own bikes and B is the students who own skateboards, find

• A∩B and P(A ∩B). Are the events independent?• AUB and P(AUB)• (AUB)’ and P(AUB)’

20

Ex. Amongst a group of 20 students, 7 are taking Math and of these 3 are also taking Biology. 5 are taking neither. What is the probability that a student chosen at random is taking Biology?Solution:

The “eggs” show MathM B

5

4 3

The diagram shows the 20 students.

and Biology

3 do both

5 do neither

7 take Math ( but we have 3 already )

20

Amongst a group of 20 students, 7 are taking Maths and of these 3 are also taking Biology. 5 are taking neither. What is the probability that a student chosen at random is taking Biology?

Solution:

M B

5

4 3 8

The diagram shows the 20 students.

The final number (taking Biology but not Math ) is given by

20 4 3 5 8

So, P( student takes Biology ) =20

11

20

83

Filling in a Venn diagram100 people are asked if they eat meat, fish, both, or neither. You are told that 55 eat meat, 52 eat fish, and 21 eat neither. Use this information to complete the Venn diagram below.

21 eat neither

28

21

27 24

Finding probabilities

Use the Venn diagram to find the probability that someone picked at random:

a) eats meat,

b) eats fish,

c) eats neither,

d) eats only fish,

e) eats both.

‘Given that’

Given that a man eats meat, find the probability that he also eats fish.

Summary of methods

The “or” ruleThe “or” rule

P(A or B) = P(AUB) = P(A) + P(B) – P(A∩B)

The word “or” often indicates that the probabilities need to be added together.

The “and” ruleThe “and” rule

P(A and B) = P(A∩B) = P(A) × P(B) if the two events are independent.

The word “and” often indicates that the probabilities need to be multiplied together.