Statistics 510: Notes 5

Statistics 510: Notes 5

Reading: Sections 2.5, 3.1-3.2

Next week’s homework will be e-mailed and posted on the web site www-stat-wharton.upenn.edu/~dsmall/stat510-f05 by tonight.

I. Application of Combinatorics to Computing Probabilities in Sample Spaces with Equally Likely Outcomes

For a sample space with equally likely outcomes, for any event ,

Example 7: A committee of 5 is to be selected from a group of 6 men and 9 women. If the selection is made randomly, what is the probability that the committee consists of 3 men and 2 women.

Note: The committee being randomly selected means that each of the possible combinations is equally likely to be selected.

Example 8: An urn contains n balls, of which one is special. If k of these balls are withdrawn one at a time, with each selection being equally likely to be any of the balls that remain at the time, what is the probability that the special ball is chosen?

Example 9: A poker hand consists of 5 cards. If the cards have distinct consecutive values and are not all of the same suit, we say that the hand is a straight. For instance, a hand consisting of the five of spades, six of spades, seven of spades, eight of spades and nine of hearts is a straight? What is the probability that one is dealt a straight?

Example 10: A deck of 52 playing cards is well shuffled and the cards turned up one at a time until the first ace appears. Is the next card – that is, the card following the first ace – more likely to be the ace of spades or the two of clubs?

Example 11: A football team consists of 20 offensive and 20 defensive players. The players are to be paired in groups of 2 for the purpose of determining roommates. If the pairing is done at random, what is the probability that there are no offensive-defensive roommate pairs? What is the probability that there are 2i offensive-defensive roommate pairs, i=1,2,...,10?

II. Conditional Probability (Chapter 3.2)

If we are trying to compute the probability of an event E and we know that an event F has occurred, we should use the information that F has occurred in computing the probability of E.

The conditional probability represents the proportion of times that E occurs in those experiments in which F has occurred in many repetitions of the experiment.

Example 1: Digitalis therapy is often beneficial to patients who have suffered congestive heart failure, but there is the risk of digitalis intoxication, a serious side effect, that is moreover difficult to diagnose. To improve the chances of a correct diagnosis, the concentration of digitalis in the blood can be measured. Let

T + = patient has high blood concentration of digitalis (positive test)

T - = patient has low blood concentration of digitalis (negative test)

D + = patient will suffer digitalis intoxication if treated with digitalis thereapy (disease present)

D - = patient will not suffer digitalis intoxication if treated with digitalis therapy (disease absent)

The probability of a randomly selected patient having each possible combination of D,T are the following:

The probability that a random selected patient has D+ (so that digitalis therapy should not be used) is 0.185+0.133 = 0.318.

What is the probability that a randomly selected patient has D+ given that the randomly selected patient has a positive test, T+?

The proportion of patients with T+ who have D+ is

.

What if the test is negative? The proportion of patients with T- who have D+ is

.

General definition of conditional probability:

If , then

.

Idea behind definition: Given that F has occurred, the relevant space becomes F rather than S. The conditional probability is a probability function on F that represents the proportion of times that E occurs in those experiments in which F has occurred in many repetitions of the experiment.

Example 2: A couple has two children. What is the probability that both are girls given that the oldest child is a girl? What is the probability that both are girls given that one of them is a girl? Assume that the four possible outcomes -- (younger is boy, older is girl), (younger is boy, older is girl), (younger is girl, older is girl), (younger is girl, older is boy) – are equally likely.

Multiplication Rule: From the definition of conditional probability, we have that

.

This is a useful formula for computing when is easy to compute.

For three events , by thinking of as a single event – say, D – we can write

Repeating the same argument for n events, we have the multiplication rule:

Example 3: An urn contains 5 white chips, 4 black chips and 3 red chips. Four chips are drawn sequentially and without replacement. What is the probability of obtaining the sequence (white, red, white, black)?

Example 4: People v. Collins

In 1964, a woman shopping in Los Angeles had her purse snatched by a young, blond female wearing a ponytail. The thief fled on foot but was seen shortly thereafter getting into a yellow automobile drive by a black male who had a mustache and a beard. A police investigation subsequently turned up a suspect, one Janet Collins, who was blond, wore a ponytail and associated with a black male who drove a yellow car and had a mustache. An arrest was made.

Not having any tangible evidence, and no reliable witnesses, the prosecutor sought to build his case on the unlikelihood of Ms. Collins and her companion sharing these characteristics and not being the guilty parties. First, the bits of evidence that were available were assigned probabilities. It was estimated, for example, that the probability of a female wearing a ponytail in Los Angeles was 1/10. The following are the probabilities quoted for the six “facts” agreed on by the victim and eyewitnesses:

The prosecutor multiplied these numbers together and claimed that the product

or 1 in 12 million was the probability of the intersection – that is, the probability that a random couple would fit this description. A probability of 1 in 12 million is so small, he argued, that the only reasonable decision is to find the defendants guilty. The jury agreed, and handed down a verdict of second-degree robbery. Later though, the Supreme Court of California disagreed. Ruling on an appeal, the higher court reversed the decision, claiming that the probability argument was incorrect and misleading.

What do you think?

Example 5: Three prisoner’s problem.

Three prisoners, A, B and C are on death row. The governor decides to pardon one of the three and chooses at random the prisoner to pardon. He informs the warden of his choice but requests that the name be kept secret for a few days.

The next day, A tries to get the warden to tell him who had been pardoned. The warden refuses. A then asks which of B or C will be executed. The warden thinks for a while, then tells A that B is to be executed.

Warden’s reasoning: Each prisoner has a 1/3 chance of being pardoned. Clearly, either B or C must be executed, so I have given A no information about whether A will be pardoned.

A’s reasoning: Given that B will be executed, then either A or C will be pardoned. My chance of being pardoned has risen to 1/2.

Which reasoning is correct?