Ch. 6 Day 5 AP Statistics

Slide 1: 

The Practice of Statistics Third Edition Chapter 6: Probability and Simulation: The Study of Randomness Copyright © 2008 by W. H. Freeman & Company

Independence and Multiplication Rule : 

Independence and Multiplication Rule The addition rule is for disjoint events Probability that one or the other event, A or B occurs. Both events cannot occur We will today describe the probability that both events A & B occur We will get to more general rules in the next section.

Tossing a Coin : 

Tossing a Coin Event A = First toss is a head. Event B = Second toss is a head These are not disjoint events. A & B can occur. We want to compute the probability of both A & B occur (that is both tosses are heads)

Slide 4: 

Both events occurring is the overlap area of the Venn Diagram. We know P(A) = .5 and P(B) = .5 What is P(A and B) Common Sense tells us P(A and B) = .5 * .5 = .25 This assumes the second toss has probability of .5 after the toss of the first coin is Heads. Event A = 1st Toss is heads. Event B = 2nd Toss is heads

Slide 5: 

For this to be true, we have to be able to say that the events are independent. Independence means that the outcome of the next event is not influenced by the outcome of the previous event. In practice, independence is assumed when we want to describe random phenomena that seem to be physically unrelated to each other.

Independent Or Not Independent : 

Independent Or Not Independent Coin tosses are independent Next toss is not influenced by last toss. Drawing a card can be independent You put the card back in the deck after you draw. Called replacement Drawing a card can be not independent. You do not put the card back in the deck after you draw.

CAUTION : 

CAUTION THE MULTIPLICATION RULE APPLIES ONLY TO INDEPENDENT EVENTS. YOU CANNOT USE IF THE EVENTS ARE NOT INDEPENDENT

Sudden Infant Death Syndrome : 

Sudden Infant Death Syndrome Causes babies to die suddenly (often in crib), but no cause is known. Greatly reduced by putting infants on their backs. When SIDS occurs more than once in the same family, parents are sometimes accused. The rate of SIDS in a non-smoking, middle class family is 1/8,500. So P(two deaths) = (1/8,500)*(1/8,500) = 1/72,250,000

What Does This Assume???? : 

What Does This Assume???? SIDS deaths are independent events. But the cause is unknown There may be environmental or genetic factors that predispose some families to SIDS. This makes a second case more likely. Has to be independent events to use Multiplication Rule

Cautions!!!! : 

Cautions!!!! Addition Rule P(A or B) = P(A) + P(B) If A and B are disjoint Multiplication Rule P(A and B) = P(A) * P(B) If A and B are independent Disjoint Events cannot be independent If A occurs, then B cannot occur.

More to Consider : 

More to Consider If two events A and B are independent Their complements are also independent Ac and Bc are independent Ac is independent of B Assume 75% of voters are Republican in a city. Interview two voters Probability first is Republican and second is not (.75)*(.25) = .1875

Six Degrees of Separation : 

Six Degrees of Separation Is this possible on-line Probability of message being passed along is .37. What is the chance of this happening? 0.375 = .0069 .69% 384 out of 24,163 – 1.58% Very unlikely to occur

A Very Good Use : 

A Very Good Use Rapid HIV Test .004 probability of a false positive (fp) If 200 people are tested, what is the probability that at least one false positive occurs? Tests are independent P(at least one fp) = 1 – P(no false positives) = 1 – P(200 negatives) = 1 – 0.996200 = 1 - .4486 = .5514 The probability is greater than .5 that there is at least one false positive.

Assignment : 

Assignment Exercises 6.47, 6.49, 6.56, 6.59, 6.61, 6.63 Read pages 435 – 440 Exam Monday, December 7th.