Some Rules of Probability

More formulae: P(B|A) = = Thus, P(B|A) is not the same as P(A|B). P(A  B) = P(A|B)·P(B) P(A  B) = P(B|A)·P(A) CONDITIONAL PROBABILITY

AIDS Testing Example  ELISA test: + : HIV positive - : HIV negative Correctness: 99% on HIV positive person (1% false negative) 95% on HIV negative person (5% false alarm)  Mandatory ELISA testing for people applying for marriage licenses in MA. “low risk” population: 1 in 500 HIV positive  Suppose a person got ELISA = +. Q: HIV positive?

Bayes’ Theorem …some people make a living out of this formula Try Michael Birnbaum’s (former UIUC psych faculty) Bayesian calculator

Bayes’ Theorem

The Binomial Distribution Today:

We have already talked about the most famous continuous random variable, which, because it is so heavily used, even has a name: The Normal Random Variable (and, associated with it, the Normal Distribution ) Today we will talk about a famous discrete random variable, which, because it is so heavily used, also has a name: The Binomial Random Variable (and, associated with it, the Binomial Distribution )

FAIR COIN “POPULATION” THEORETICAL PROBABILITY OF HEADS IS ½ If you toss a fair coin 10 times, what is the probability of x many heads?

Binomial Random Variable Potential Examples: Repeat a simple dichotomous experiment n times and count… Coin Tossing:# heads Marketing:# purchases Medical procedure:# patients cured Finance:# days stock  Politics:# voters favoring a given bill Testing:# number of test items of a given difficulty that you solve correctly Sampling:# of females in a random sample of people Education:# of high school students who drink alcohol

Population Repeat simple experiment n many times independently.

X: number of successes in n many independent repetitions of an experiment, each repetition having a probability p of success Binomial Random Variable

The Binomial Distribution

Trial Trial Trial Trial Probab. (letting q=1-p) pppppppp pppqpppq ppqpppqp ppqqppqq pqpppqpp pqpqpqpq pqqppqqp pqqqpqqq qpppqppp qppqqppq qpqpqpqp qpqqqpqq qqppqqpp qqpqqqpq qqqpqqqp qqqqqqqq X

Probab. (letting q=1-p) pppppppp pppqpppq ppqpppqp ppqqppqq pqpppqpp pqpqpqpq pqqppqqp pqqqpqqq qpppqppp qppqqppq qpqpqpqp qpqqqpqq qqppqqpp qqpqqqpq qqqpqqqp qqqqqqqq X

In general, for n many trials How do we figure that out in general?

Factorial

Binomial Distribution Formula: TABLE C Pages T-6 to T-10 in the book

Probab. (letting q=1-p) pppppppp pppqpppq ppqpppqp ppqqppqq pqpppqpp pqpqpqpq pqqppqqp pqqqpqqq qpppqppp qppqqppq qpqpqpqp qpqqqpqq qqppqqpp qqpqqqpq qqqpqqqp qqqqqqqq X

BINOMIAL COEFFICIENTS:

Example: # heads in 5 tosses of a coin: X~B(5,1/2) Expectation Variance # heads in 5 tosses of a coin:

PROOF :

Another Statistic: The Sample Proportion Remember that X is a random variable The sample proportion is a linear transformation of X and thus a random variable too Sampling Distribution of the Sample Proportion:

The Sample Proportion Unbiased Estimator

Let’s take another look at some Binomial Distributions especially what happens as n gets bigger and bigger

Normal Approximation of/to the Binomial

Normal Approximation

Normal Approximation: Let’s check it out! TABLE C Pages T-6 to T-10 in the book

Normal Approximation: Let’s check it out! Standardizing:

Normal Approximation: Let’s check it out!

Are we stuck with a bad approximation??

Binomial Distribution, n= # of successes probability 0.2

For now, that’s it … We will revisit the Binomial: Based on the sample proportion as an estimate of the population proportion, we will develop “confidence intervals” for the population proportion. We will carry out hypothesis tests about the true population proportion, using the information gained from the sample proportion.