STAT 104: Section 4

18 Oct, 2007TF: Daniel Moon

Agenda of Today Observational Study vs. Experimental of

Design Probability Rules Bayes’ Rule Random Variables

Observational Study vs. Experimental Design

Collects information from individuals making no attempt to influence the responses

Various ways to collect information Simple random

sampling, stratified random sampling

Imposes an intervention on individuals in order to observe their responses

3 principles Control Randomization Repetition

Observational StudyObservational Study Experimental DesignExperimental Design

Probability Rules

Basic Formulas

P(A and B) = P(A) X P(B|A) Independence

Example of Conditional Probability

Let’s use random variables next time!

Find the iPod!!!

What is the probability you picked up friend A’s iPod given the first song you listened was Rock?

• Friend A listens to, 30% Rock, 70% Ballads

• Friend B listens to, 80% Rock, 20% Ballads

• Given a 50% chance you picked up friend A or friend B’s ipods

Examples The clumsy jewel thief. A jewel thief and his partner planned to steal 2 identical

diamonds from a jewelry store. They had 2 similar, but fake diamonds prepared.

The plan was for the thief’s accomplice to faint and, while the store’s staff were distracted, for the thief to grab the two real diamonds from the viewing pad and replace them with the two fake diamonds.

On the day of the robbery all was going according to plan up to the point when the thief’s accomplice fainted and the staff rushed to her assistance. The thief grabbed the two real diamonds from the viewing pad and put them in his pocket, but then to his horror realized that the two fake diamonds were already in that same pocket.

The thief had to quickly grab two stones at random from the four (two real and two fake) in his pocket and leave them on the viewing pad after which he left the store with his accomplice.

Examples Give the sample space S that

corresponds to the two stones in the thief’s pocket as he leaves the store [(D1,F1), etc.] and assign probabilities to the elements of that sample space.

Examples Let X be the number of real diamonds the

thief had in his pocket as he left the store. Give the probability distribution of X.

What is the probability that at least one of the two stones in the thief’s pocket is real?

Examples What is the expected number of real diamonds in

the thief’s pocket (show your work).

The thief takes the 2 stones to an appraiser to find out how many real stones he has. The thief hands a stone to the appraiser and after looking at the stone under a magnifying lens, the appraiser says it is a real diamond. Knowing that at least one of the stones is a real diamond, what is the probability that the second stone, still in the thief’s pocket, is also real?

Binomial Probability

Examples 2. Nationwide it is known that 20% of Dell laptop

computers will fail in the first year (failure of the power supply or disk drive, etc.).

Three Harvard Freshmen men are roommates and have each (by chance) purchased a new Dell laptop computer. What is the probability that at least one of these three Dell laptops will fail during the year?

Examples 2. One thousand Harvard students purchased a

new Dell laptop this year. What is the probability that more than 220 of these 1,000 Dell laptops will fail in the first year?