Probability rules

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Probability Units

We can use different rules to find the probability of an event.

Rule 1:
Pr( A ) + Pr( A' ) = 1
Where A is the event, A' is the complement.

Rule 2:
For mutually exclusive events (disjoint events):

A

B

Rule 3 :
For non-exclusive events (joint events) :

Insert a diagram A and B joint events

Rule 4 :
For indepenent events:

Examples

Example 1:
A fruit bowl contains 5 bananas, 4 apples, 1 orange and 2 peaches. If I choose a fruit from the bowl, find:
(a) Pr( choosing a banana)
(b) Pr( choosing a peach or an apple )
(c) Pr(not choosing an apple)

Let's take B for bananas, number of bananas n(B) = 5
               A for apples, number of apples n(A) = 4
               O for orange, number of oranges n(O) = 1
               P for peaches, number of peaches n(P) = 2

The total number of fruits in the fruit bowl is 12. n(S) = 12
So, the Pr( choosing a banana) = Pr(B)

Have a Go

Problem 1
A bag contains 12 red marbles and 8 black marbles.Two marbles are taken out in succession.Their colours are recorded and neither is replaced. What is the probability:
(a) that the both marbles will be red?
(b) that the both marbles will be either red or black?

Problem 2
A bag contains 12 red marbles and 8 black marbles.Two marbles are taken out in succession.Their colours are recorded and the marbles are replaced into the bag.
What is the probability:
(a) that the both marbles will be red?
(b) that the both marbles will be either red or black?

Practice Questions

Question 1
A green die and a red die are tossed. Let's define the events, X={2 with the green die} and Y= {5 with the red die}.
(a) Find the Pr(X)

(b) Find the Pr(Y)

(d) Are X and Y independent events?

Click to See Answer

Question 2
A card is drawn from a normal pack of 52 playing cards. Find the probability that the card is:

(a) a black card

(b) not a club

Click to See Answer

Question 3
A bag contains 3 marbles, coloured green,red and blue. A marble is taken out and the colour was recorded,then replaced in the bag before a second selection is made.

(a) Find the Pr(two green marbles)

(b) Find the Pr(selecting different coloured marbles)

Click to See Answer

Solution 1
A bag contains 12 red marbles and 8 black marbles.Two marbles are taken out in succession.Their colours are recorded and neither is replaced. What is the probability:
(a) that the both marbles will be red?
(b) that the both marbles will be either red or black?

In this problem, once the marble is taken out it is not placed back in the bag. This process is called sampling without replacement.

At the beginning there are 20 marbles in the bag.
Let's take R ={red marbles}, n(R) = 12 and B = {black marbles}, n(B) = 8
Total number of marbles n(S) = 20
This problem can be shown in a probability tree diagram.

=0.35.

.=0.25

=0.25

=0.15

back to Have a Go

Solution 2
A bag contains 12 red marbles and 8 black marbles.Two marbles are taken out in succession.Their colours are recorded and the marbles are replaced into the bag.
What is the probability:
(a) that the both marbles will be red?
(b) that the both marbles will be either red or black?

In this problem, once the marble is taken out and recorded it's colour,it is placed back in the bag. This process is called sampling with replacement.

This problem can be shown in a probability tree diagram.

=0.36

=0.24

=0.24

=0.16

back to Have a Go

Answer to Question 1

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Answer to Question 2

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Answer to Question 3

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