Download Twenty-sixth Lecture on March 25

Section 7.4 Use of counting Techniques in Probability
Computing the probability of an event in a uniform
sample space:
Let S be a uniform sample space and let E be any event.
Then,
of favorable outcomes in E
n( E )
P (E) = Number

Number of favorable outcomes in S
n( S )
Example 1
A fair coin is tossed six times
1. What is the probability that the coin will land
heads exactly three times?
2. What is probability that the coin will land heads at
most three times?
3. What is the probability that the coin will land
heads on the first and the last toss?
Example 2:
Two cards are selected at random from well-shuffled
pack of 52 playing cards.
1. What is the probability that they are both aces?
2. What is the probability that neither of them is ace?
Example 3:
A group of 5 people is selected at random. What is the
probability that at least two of them have same
birthday?
Example 4:
An exam consists of ten true-false questions. If a student
guesses at every answer, what is the probability that
he/she will answer exactly six questions correctly?
Section 7.5 Conditional Probability and Independent
Events
Conditional Probability of an event:
If A and B are events in experiment and P(A)  0, then
the conditional probability that the event B will occur
given that the event A has already occurred is
P(B|A) = P(PA(A)B)
The Product rule:
P( A  B)  P( A) * P( B | A)
P( A  B  C)  P( A) * P( B | A) * P(C | A  B)
Example 1:
In a test recently conducted by the U.S. army, it was
found that 1000 new recruits, 600 men and 400 women,
50 of the men and 4 of the women were red-green
color-blind. Given that a recruit selected at a random
from this group is red-green color –blind, what is the
probability that the recruit is male? What is the
probability that a recruit selected from this group is
women who does not have red-green color-blind.
Example 2:
Two cards are drawn without replacement from a wellshuffled deck of 52 playing cards. What is the
probability that the first card drawn is an ace and the
second card drawn is face card?
Example 3
Suppose that a box contains two defective Charismas
tree lights that have been inadvertently mixed with
eight non-defective lights. If the lights are selected on at
a time without replacement and tested until both
defective lights are found, what is the probability that
both defective lights will be found after three trails?
Test for the independence of two events
Two events A and B are independent if and only if
P( A  B)  P( A) * P( B)
Example 4:
Consider the experiment consisting of tossing a fair coin
twice and observing the outcomes. Show that the event
of ‘head’ in the first toss and ‘tail’ in the second toss are
independent events.
Example 5
If it is estimated that 0.8% of large consignment of eggs
in a certain supermarket is broken.
1. What is the probability that a customer who
randomly selected a dozen of these eggs receives at
least one broken egg?
2. What is the probability that a customer who selects
these eggs at random will have to check three
cartons before finding a carton without any broken
eggs? (Each carton contains a dozen eggs.)