Mutually exclusive events cannot occur at the same time. The probability of event A or B occurring is calculated as P(A) + P(B). Independent events do not influence each other, so P(A and B) = P(A) × P(B). Dependent events influence each other, so P(A and B) = P(A) × P(B after A). Examples are provided to demonstrate calculating probabilities for mutually exclusive, independent, and dependent events.

1. MUTUALLY EXCLUSIVE EVENTS

2. • Events are mutually exclusive if they
cannot happen at the same time. For
example, if we toss a coin, either heads
or tails might turn up, but not heads
and tails at the same time. Similarly, in
a single throw of a die, we can only
have one number shown at the top
face. The numbers on the face are
mutually exclusive events

3. • If A and B are mutually exclusive
events then the probability of A
happening OR the probability of B
happening is P(A) + P(B).
• P(A or B) = P(A) + P(B)

4. Example 1
• What is the probability of a die showing a 2 or
a 5?

5. Practice
• The probabilities of three teams A, B and C
winning a badminton competition are
• Calculate the probability that
• a) either A or B will win
• b) either A or B or C will win
• c) none of these teams will win
• d) neither A nor B will win

6. Solution/s
c) P(none will win) = 1 – P(A or B or C will win)
d) P(neither A nor B will win) = 1 – P(either A or B will win)

7. Independent Events
• Events are independent if the outcome of
one event does not affect the outcome of
another. For example, if you throw a die
and a coin, the number on the die does
not affect whether the result you get on
the coin.
• If A and B are independent events, then
the probability of A happening AND the
probability of B happening is P(A) × P(B).
• P(A and B) = P(A) × P(B)

8. Example 1
• If a dice is thrown twice, find the probability
of getting two 5’s.

9. Two sets of cards with a letter on each card
as follows are placed into separate bags.
Sara randomly picked one card from each bag.
Find the probability that:
a) She picked the letters ‘J’ and ‘R’.
b) Both letters are ‘L’.
c) Both letters are vowels.

10. Solution for no. 2
a) Probability that she picked J and R
=
b) Probability that both letters are L =
c) Probability that both letters are
vowels =

11. Example 3
• Two fair dice, one colored white and
one colored red, are thrown. Find
the probability that:
• a) the score on the red die is 2 and
white die is 5.
• b) the score on the white die is 1 and
red die is even

12. Solution for No. 3
a) Probability the red die shows 2
and white die 5 =
b) Probability the white die shows 1
and red die shows an even
number =

13. DEPENDENT EVENTS
• Events are dependent if the outcome of
one event affects the outcome of
another. For example, if you draw two
colored balls from a bag and the first
ball is not replaced before you draw the
second ball then the outcome of the
second draw will be affected by the
outcome of the first draw.

14. • If A and B are dependent events,
then the probability of A happening
AND the probability of B happening,
given A, is P(A) × P(B after A).
• P(A and B) = P(A) × P(B after A)
• P(B after A) can also be written as
P(B | A)
• then P(A and B) = P(A) × P(B | A)

15. Example 1
• A purse contains four P50
bills, five P100 bills and
three P20 bills. Two bills are
selected without the first
selection being replaced.
Find P(P50, then P50)

16. • There are four P50 bills.
• There are a total of twelve bills.
• P(P50) = 4/12
• The result of the first draw affected
the probability of the second draw.
• There are three P50 bills left.
• There are a total of eleven bills left.
• P(P50 after P50) = 3/11

17. • P(P50, then P50) = P(P50) · P(P50
after P50) = (4/12)x(3/11)=12/132
• The probability of drawing a P50
bill and then a P50bill is

18. Dependent:Practice
• A bag contains 6 red, 5 blue and 4
yellow marbles. Two marbles are
drawn, but the first marble drawn
is not replaced.
• a) Find P(red, then blue).
• b) Find P(blue, then blue)

19. Independent Events: Practice
• Two fair dice, one colored white and
one colored red, are thrown. Find
the probability that:
• a) the score on the red die is 2 and
white die is 5.
• b) the score on the white die is 1 and
red die is even

20. Mutually ExclusiveEvents:Practice
• The probabilities of three teams A, B and C
winning a badminton competition are
• Calculate the probability that
• a) either A or B will win
• b) either A or B or C will win
• c) none of these teams will win
• d) neither A nor B will win