PROBABILITY AND STATISTICS

WEEK 3

Onur Doğan

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 ProbabilitySuppose 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 ProbabilityA 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 ProbabilityLottery 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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ExampleIt 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;

•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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Test Results

Negative Positive (B)

Corrupted (A) 10 40

Not corrupted 45 5

The Multiplication Rule

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ExampleSelecting 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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ExampleA 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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