2. Chapter 11: Further Topics in Algebra
11.1 Sequences and Series
11.2 Arithmetic Sequences and Series
11.3 Geometric Sequences and Series
11.4 Counting Theory
11.5 The Binomial Theorem
11.6 Mathematical Induction
11.7 Probability

3. 11.7 Probability Basic Concepts
An experiment has one or more outcomes. The outcome of rolling a die is a number from 1 to 6.
The sample space is the set of all possible outcomes for an experiment. The sample space for a dice roll is {1, 2, 3, 4, 5, 6}.
Any subset of the sample space is called an event. The event of rolling an even number with one roll of a die is {2, 4, 6}.

4. 11.7 Probability Probability of an Event E
In a sample space with equally likely outcomes, the probability of an event E, written P(E), is the ratio of the number of outcomes in sample space S that belong to E, denoted n(E), to the total number of outcomes in sample space S, denoted n(S). That is,

5. 11.7 Finding Probabilities of Events
Example A single die is rolled. Give the probability
of each event.
(a) E3 : the number showing is even
(b) E4 : the number showing is greater than 4
(c) E5 : the number showing is less than 7
(d) E6 : the number showing is 7

6. 11.7 Finding Probabilities of Events
Solution The sample space S is {1, 2, 3, 4, 5, 6} so
n(S) = 6.
(a) E3 = {2, 4, 6} so
(b) E4= {5, 6} so

7. 11.7 Finding Probabilities of Events
Solution
(c) E5 = {1, 2, 3, 4, 5, 6} so
(b) E6 = Ø so

8. 11.7 Probability For an event E, P(E) is between 0 and 1 inclusive.
An event that is certain to occur always has probability 1.
The probability of an impossible event is always 0.

9. 11.7 Complements and Venn Diagrams
The set of all outcomes in a sample space that do not belong to event E is called the complement of E, written E´. If S = {1, 2, 3, 4, 5, 6} and E = {2, 4, 6} then E´ = {1, 3, 5}.

10. 11.7 Complements and Venn Diagrams
Probability concepts can be illustrated with Venn diagrams. The rectangle represents the sample space in an experiment. The area inside the circle represents event E; and the area inside the rectangle but outside the circle, represents event E´.

11. 11.7 Using the Complement Example A card is drawn from a well-shuffled
Deck. Find the probability of event E, “the card is
an ace,” and event E´.
Solution There are 4 aces in the deck of 52
cards and 48 cards that are not aces. Therefore,

12. 11.7 Odds The odds in favor of an event E are expressed as the
ratio of P(E) to P(E´) or as the fraction

13. 11.7 Finding Odds in Favor of an Event
Example A shirt is selected at random from a dark
closet containing 6 blue shirts and 4 shirts that are
not blue. Find the odds in favor of a blue shirt
being selected.
Solution E is the event “blue shirt is selected”.

14. 11.7 Finding Odds in Favor of an Event
Solution The odds in favor of a blue shirt are
or 3 to 2.

15. 11.7 Probability Probability of the Union of Two Events
For any events E and F,

16. 11.7 Finding Probabilities of Unions
Example One card is drawn from a well-shuffled
deck of 52 cards. What is the probability of each
event?
(a) The card is an ace or a spade.
(b) The card is a 3 or a king.

17. 11.7 Finding Probabilities of Unions
Solution (a) P(ace or spade) = P(ace) + P(spade)
– P(ace and spade)
(b) P(3 or K) = P(3) + P(K) – P(3 and K)

18. 11.7 Probability Properties of Probability For any events E and F, 1.
2. P(a certain event) = 1;
3. P(an impossible event) = 0; 4. 5.

19. 11.7 Binomial Probability An experiment that consists of
repeated independent trials,
only two outcomes, success and failure, in
each trial,
is called a binomial experiment.
Let the probability of success in one trial be p.
Then the probability of failure is 1 – p.
The probability of r successes in n trials is given by

20. 11.7 Finding Binomial Probabilities
Example An experiment consists of rolling a die 10
times. Find the probability that exactly 4 tosses result
in a 3.
Solution Here , n = 10 and r = 4. The required probability is