Transcript Week 4
Example
Suppose that a deck of 52 cards containing
four aces is shuffled thoroughly and the cards
are then distributed among four players so that
each player receives 13 cards. We shall
determine the probability that each player will
receive one ace.
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Example
Suppose that a fair coin is to be tossed 7 times,
and it is desired to determine;
(a) the probability p of obtaining three
heads
(b) the probability p of obtaining three or
fewer heads.
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Binomial Coefficients
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Multinomial Coefficients
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Multinomial Coefficients
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Example
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PROBABILITY AND
STATISTICS
WEEK 4
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Conditional Probability
A major use of probability in statistical inference
is the updating of probabilities when certain
events are observed. The updated probability of
event A after we learn that event B has occurred
is the conditional probability of A given B.
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Conditional Probability
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Conditional Probability
Suppose that of all individuals buying a certain digital
camera, 60% include an optional memory card in their
purchase, 40% include an extra battery, and 30% include
both a card and battery. Consider randomly selecting a
buyer and let A {memory card purchased} and B {battery
purchased}.
• What’s the probability that selected individual purchased
an extra battery is also purchased an optional card?
• What’s the probability that selected individual purchased
optional card is also purchased an extra battery?
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Conditional Probability
Rolling Dice. Suppose that two dice were rolled and
it was observed that the sum T of the two numbers
was odd. We shall determine the probability that T
was less than 8.
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Conditional Probability
A news magazine publishes three columns entitled “Art” (A),
“Books” (B), and “Cinema” (C). Reading habits of a
randomly selected reader with respect to these columns are;
•What’s the probability that a selected reader, who reads books also reads arts?
•If it is known that selected person reads at least one column, what’s the probability
that he reads art?
•If it is known that the selected person reads cinema, what’s the probability that he
reads book and art also?
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Conditional Probability
Lottery Ticket. Consider a state lottery game in which six numbers
are drawn without replacement from a bin containing the numbers
1–30. Each player tries to match the set of six numbers that will be
drawn without regard to the order in which the numbers are drawn.
Suppose that you hold a ticket in such a lottery with the numbers 1,
14, 15, 20, 23, and 27. You turn on your television to watch the
drawing but all you see is one number, 15, being drawn when the
power suddenly goes off in your house. You don’t even know
whether 15 was the first, last, or some in-between draw. However,
now that you know that 15 appears in the winning draw, the
probability that your ticket is a winner must be higher than it was
before you saw the draw. How do you calculate the revised
probability?
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Independence
• Two events A and B are stochastically independent if
the occurrence of A does not affect the probability of
B. In other words, two events A and B are
independent if and only if;
P(A /B) = P(A)
P(B / A) = P(B)
P(A B) = P(A).P(B)
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Example
It is known that in a laboratory 50 computers have some kind of
virus and 50 computers have not. And a virus program have been
operated;
Test Results
Negative
Positive (B)
Corrupted (A)
10
40
Not corrupted
45
5
•Find the probability that virus test is positive when a computer has virus?
•Find the probability that virus test is positive when a computer has no virus?
• Check if events A and B are independent.
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The Multiplication Rule
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Example
Selecting Two Balls. Suppose that two balls are to be
selected at random, without replacement, from a box
containing r red balls and b blue balls. We shall
determine the probability p that the first ball will be
red and the second ball will be blue.
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The Multiplication Rule for
Conditional Probabilities
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The Multiplication Rule
Selecting Four Balls. Suppose that four balls
are selected one at a time, without
replacement, from a box containing r red balls
and b blue balls (r ≥ 2, b ≥ 2). We shall
determine the probability of obtaining the
sequence of outcomes red, blue, red, blue.
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Example
A chain of video stores sells three different brands of DVD players. Of its DVD player
sales, 50% are brand 1, 30% are brand 2, and 20% are brand 3. Each manufacturer
offers a 1-year warranty on parts and labor. It is known that 25% of brand 1’s DVD
players require warranty repair work, whereas the corresponding percentages for brands
2 and 3 are 20% and 10%, respectively.
•What is the probability that a randomly selected purchaser has bought a brand 1 DVD
player that will need repair while under warranty?
•What is the probability that a randomly selected purchaser has a DVD player that will
need repair while under warranty?
•If a customer returns to the store with a DVD player that needs warranty repair work,
what is the probability that it is a brand 1 DVD player? A brand 2 DVD player? A brand
3 DVD player?
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The Law of Total Probability
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Example
Two boxes contain long bolts and short bolts.
Suppose that one box contains 60 long bolts
and 40 short bolts, and that the other box
contains 10 long bolts and 20 short bolts.
Suppose also that one box is selected at
random and a bolt is then selected at random
from that box. We would like to determine the
probability that this bolt is long.
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Bayes’ Theorem
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Example
For the previous question;
If we know the selected bolt is long, determine
the probability that the bolt come from first
box?
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Example
Three different machines M1, M2, and M3 were used for
producing some items. Suppose that 20 percent of the items
were produced by machine M1, 30 percent by machine M2,
and 50 percent by machine M3. Suppose further that 1
percent of the items produced by machine M1 are defective,
that 2 percent of the items produced by machine M2 are
defective, and that 3 percent of the items produced by
machine M3 are defective. Finally, suppose that one item is
selected at random from the entire batch and it is found to be
defective. We shall determine the probability that this item
was produced by machine M2.
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Example
Customers are used to evaluate preliminary product designs.
In the past, 95% of highly successful products, 60% of
moderately successful products, and 10% of poor products
received good reviews. In addition, 40% of product designs
have been highly successful, 35% have been moderately
successful, and 25% have been poor products.
• Find the probability that a product receives a good review.
• What is the probability that a new design will be highly
successful if it receives a good review?
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