Independent Events
Examples of independent events
1. You flip a coin and get a heads and you flip a
second coin and get a tails. The two coins don't
influence each other.
2. The probability of rain today and the probability of
a pop quiz. Quizzes happen, rain or shine.
3. The probability of a having a car crash in the next
25 000Km’s is independent of the number of car
crashes you’ve had in the last 25 000 Km’s.
Examples of non independent events
1. You draw one card from a deck and its black and you
draw a second card and its black. By removing one black
card, you made the probability of drawing a second one
slightly smaller. Technically this is called 'sampling without
replacement'.
2. The probability of snow today and the probability of a pop
quiz;
Snow causes school closings, in which case your teacher
can't give you a pop quiz.
3. The chance that you are hungry right now and the chance
that you're eating right now. One leads to the other
eventually ...
Mutually Exclusive Events
Unable to both happen at the same time
Mutually Exclusive Example
Consider rolling a dice. What is the probability
or rolling a 2 or a 5?
We can say rolling both is mutually exclusive. Or
the probability of rolling a two and a five is zero.
The probability of rolling a two or a five is 2/6
(1/6 + 1/6).
Mutually Exclusive Examples
Two dice are rolled. We define events E1, E2, E3 and E4 as
follows
E1: Getting a sum equal to 10
E2: Getting a double
E3: Getting a sum less than 4
E4: Getting a sum less than 7
a) Are events E1 and E2 mutually exclusive?
b) Are events E2 and E3 mutually exclusive?
c) Are events E3 and E4 mutually exclusive?
d) Are events E4 and E1 mutually exclusive?
Mutually Exclusive Examples
a) E1 and E2 are not mutually exclusive because outcome (5,5) is a double
and also gives a sum of 10. The two events may occur at the same time.
b) E2 and E3 are not mutually exclusive because outcome (1,1) is a double
and gives a sum of 2 and is less than 4. The two events E2 and E3 may occur
at the same time.
c) E3 and E4 are not mutually exclusive a sum can be less than 7 and less
than 4 a the same time. Example outcome (1,2).
d) E4 and E1 are mutually exclusive because a sum less than 7 cannot be
equal to 10 at the same time. The two events cannot occur at the same time.
Mutually Exclusive Events
Unable to both happen at the same time
Complementary Events
Complementary Examples
Consider a person being selected at random
from a class of 30. The probability a male is
selected is complementary to the probability a
female is selected.
If there were 20 males and 10 females then the
probabilities would be 2/3 and 1/3 respectively.
The probability of complementary events always
add to 1!
Complementary examples
1. Consider the probability of rain…..The
probability of no rain is complementary to
the probability of rain.
2. The probability of the Black Caps winning the
CWC is complementary with the probability
of not winning the CWC.
Match the word with the definition
Word
Definitions
Sample Space
One ‘run through’ of an
experiment
Trial
The set of all possible outcomes
Long run relative frequency
The result of a single ‘run
through’
Outcome
A good estimate of the true
probability
Theoretical Probability
The calculation of probability
from carrying out an experiment
– often called experimental
probability
Probability Notation
Let A = The event NZ wins the CWC semi final
Let B = The event SA wins the CWC semi final
P(A) =
P(A’) =
P(A n B) =
P(A u B) =