1. Probability

2. Example 1:
If a card is drawn at random
from an ordinary deck of 52
cards, find the probability that
it is a spade or an even
numbered card.

3. Example 2:
The probability that a patient entering
UMC Hospital will consult a physician is
0.7, that he/she will consult a dentist is
0.5 and that he/she will consult a
physician or a dentist or both is 0.9. What
is the probability that a patient entering
the hospital will consult both a physician
and a dentist?

4. Example 3:
Three dice are thrown,
what is the probability that
the 3 dice show even
numbers?

5. Example 4:
The probability that a certain movie will
get an award for good acting is 0.16, the
probability that it will get an award for
good directing is 0.27, and the probability
that it will get awards for both is 0.11.
What is the probability that the movie
will get only one of the two awards?

6. Example 5:
In a high school graduating class of 100
students, 54 studied mathematics, 69
studied history, and 35 studied both
mathematics and history. If one of this
students is selected at random, find the
probability that: (a) the student took
mathematics or history but not both. (b) a
student did not take either of these
subjects. (c) the student took history but
not mathematics.

7. Example 6:
There are two flocks of birds, one below
the other. The lower flock says “If one of
us goes up there, you will double our
number. But if one of you goes down
here, we will be equal in number.” If a
bird is chosen from these flocks, what is
the probability that the bird chosen at
random is from the lower flock?

8. Example 7:
In a poker hand consisting
of 5 cards, find the
probability of holding 4
hearts and 1 club.

9. Conditional Probability
The probability of an event B occurring
when it is known that some event A has
occurred is called a conditional
probability. It is denoted by the symbol
P(BІA) which is usually read as “the
probability that B occurs given that A
occurs” or simply “the probability of B
given A.”

10. Conditional Probability
Definition:
Let A and B be events such that P(A) ≠ 0.
The conditional probability of B, given A,
denoted by P(BІA), is given by
   
 AP
BAP
ABP



11. Example 1:
A card is drawn from a standard
deck. Suppose we are told that
the card picked is a spade. What
is the probability that the card
drawn is the ace of spades?

12. Example 2:
A single fair die is rolled once.
What is the probability that the
number obtained is less than 4
knowing that odd number is
rolled.

13. Example 3:
The probability that a regular scheduled
flight departs on time is P(D) = 0.83; the
probability that it arrives on time is P(A) =
0.82; and the probability that it departs
and arrives on time is P(DA) = 0.78.
Find the probability that a plane (a)
arrives on time given that it departed on
time, and (b) departed on time given that
it has arrived on time.

14. Example 4:
Consider the population of adults in small town who have
completed the requirements for a college degree. We shall
categorize them according to gender and employment
status:
One of these individuals is to be selected at random.
Consider the following events:
M: a man is chosen
E: the chosen is employed
Employed Unemployed Total
Male 460 40 500
Female 140 260 400
Total 600 300 900

15. Example 5:
Consider the population of adults in small town who have
completed the requirements for a college degree. We shall
categorize them according to gender and employment
status:
One of these individuals is to be selected at random.
Consider the following events:
F: a female is chosen
U: the chosen is unemployed
Employed Unemployed Total
Male 460 40 500
Female 140 260 400
Total 600 300 900

16. Independent Events
Two events are considered to
be independent if the
occurrence or non-occurrence
of one has no influence of the
other.

17. Independent Events
Definition:
Two events are said to be independent if
any one of the following conditions are
satisfied:
Otherwise, A and B are dependent.
     
      0if
0if


APBPBAP
BPAPABP

18. Multiplicative Rules
If events A and B are dependent,
then:
If events A and B are independent,
then
     ABPAPBAP 
     BPAPBAP 

19. Example 1:
The probability that Jack will correctly
answer the toughest question in an exam
is 1/4. The probability that Rose will
correctly answer the same question is
4/5. Find the probability that both will
answer the question correctly, assuming
that they do not copy from each other.

20. Example 2:
On example 1, find the probability
that (a) Jack will get an incorrect
answer and Rose will get the correct
answer. (b) Jack will get the correct
answer and Rose will get an incorrect
answer. (c) both will get an incorrect
answer.

21. Example 3:
Suppose that we have a fuse box
containing 20 fuses, of which 5 are
defective. If two fuses are selected at
random and removed from the box
in succession without replacing the
first, what is the probability that both
fuses are defective?

22. Example 4:
On example 3, what is the probability
that (a) the first one is not defective
and the second is defective. (b) the
first one is defective and the second
one is defective. (c) both are not
defective.

23. Exercise 1:
Three cards are drawn in succession,
without replacement, from an
ordinary deck of playing cards. Find
the probability that the first card is a
red ace, the second card is a 10 or a
jack and the third card is greater than
3 but less than 7.

24. Exercise 2:
A purse contains 3 1-peso coins and
2 5-peso coins. Another coin purse
contains 2 1-peso coins and 5 5-peso
coins . What is the probability that if
a coin is selected at random, the coin
is 5 peso? (page 135, no. 1)

25. Exercise 3:
Box 1 contains 6 good and 2 defective
light bulbs. Box 2 contains 3 good and 3
defective light bulbs. A box is selected at
random and a light bulb is chosen from it.
Find (a) the probability that the light bulb
is defective. (b) the probability that the
light bulb came from Box 2 given that it is
defective.

26. Exercise 4:
A box contains 6 red and 4 black
balls. Two balls from the box are
drawn one at a time without
replacement. What is the probability
that the second ball is red if it is
known that the first is red? (page
139, no. 11)

27. Exercise 5:
A hospital spokesperson reported that 4
births had taken place at the DLS-UMS during
the last 24 hours. Find the following
probabilities: (a) P(A) = that 2 boys and 2
girls are born. (b) P(B) = no boys are born. (c)
P(C) at least one boy is born. (d) P(AІC) (e)
P(BІC) (f) Are A and C mutually exclusive? Are
A and C independent? (g) Are B and C
mutually exclusive? Are B and C
independent? (page 145, no. 25)