Unit 6 Random Variables

Bottled vs. Tap Go to corresponding station and pick up three cups (A, B, & C) Drink all the water in Cup A first, then cup B, and finally cup C. Write down the letter of the cup that you think held bottled water DO NOT DISCUSS! What percent of the class correctly identified?

Station 1: B Station 2: A Station 3: C Let’s assume that no one can distinguish tap from bottled in our class. In that case, students would just be guessing which cup tastes different. If so, what’s the probability that an individual student would guess correctly? How many correct identifications would you need to see to be convinced that the students in your class aren’t just guessing? Design and carry out a simulation to answer this question. What do you conclude about your class’s ability to distinguish tap from bottled?

When another class did this, 13 out of 21 STUDENTS MADE CORRECT IDENTIFICATIONS. If we assume that students in his class can’t tell tap from bottled (just guessing) then they have a 1/3 chance of being correct. So we’d expect about 1/3 of the 21 students to guess correctly. How likely is it that as many as 13 of his 21 students would guess correctly? To answer this question without a simulation we need a different kind of probability model than the ones we saw in Chapter 5

6.1A Discrete Random Variables and their Probability Distributions Use a prob dist to ans Q’s about poss values of a random variable Calc mean of a discrete rand var Interpret mean in context Calc stdev of a discrete rand var

Chance Experiment One in which the outcomes are not certain—rolling a die, flipping a coin, conception of boy vs. girl, seeing if the next car at a stop sign is white……

Note the lack of modifier→ →Random Variable (denoted by X, Y, etc.) A numerical outcome of a chance process. For example, counting the number of heads out of 6 flips of a coin, counting the number of rolls until a die shows a Four, measuring the length of a randomly chosen dominant foot…. Note the lack of modifier→

Discrete Random Variable Takes on a set of numeric values which have gaps between them.

Discrete Random Variable? # of rolls of a die until a 6 appears Age of a random high school senior Value of cash in pocket/purse, etc. Gender distribution in a family of five # of students out of 10 left-handed Card configurations in poker Circumference of a random mesquite 1) yes 2) no (all depends on how we measure, floor function, culture) not just because how we report it 3) yes 4) values are categories– how can I turn it into a discrete random variable, we count the category. 5) 6) 7) no

Probability Distribution of a Discrete Random Variable X Lists the values of xi and their probabilities pi which of course must follow the requirements: 0 ⦤ pi ⦤ 1 The sum of all pi = 1

Example: (a random variable and its distribution are given to you!) In Britain, social classes are thought to be much more clearly delineated than in the U.S. The following chart represents a group of men whose fathers were deemed to be in the lowest class of British society.

Social Class (1 to 5) of Men in Britain Given Father in Class 1 (lowest class) X 1 2 3 4 5 P(X) 0.48 0.38 0.08 0.05 0.01 Is it legit? We define social mobility as just not in your same class. 1-.48 (still in same class as Dad) P(4 or 5)= P(4)+P(5)- P(Both)---which is 0 since mutually exclusive

Example: # of Courses for which a randomly chosen UA student is registered 1 2 3 4 5 6 7 P(Y) 0.02 0.03 0.09 0.25 0.40 0.16 0.05

Example: (you have to find your own probability distribution!) Suppose that 40% of all students oppose use of fees to support student interest groups and the opinions of the three students on the university advisory board are independent. S, O combination chart (8 possibilities)– use counting principles, probabilities are not the same so not #3 rule OOO,

Find the probability distribution of W. Let W = # of students out of three who would oppose using fees to support student interest groups. Find the probability distribution of W.

Find the probability distribution of X. Suppose 78% of drivers have auto insurance. Choose four drivers at random and let X = # with insurance. Find the probability distribution of X.

6.1B Discrete Random Variables and their Probability Distributions Calc mean of a discrete rand var Interpret mean in context Calc stdev of a discrete rand var

Roulette Finding the mean of a discrete random variable American roulette wheel: 38 slots numbered 1-36, plus 0, and 00. Half of the slots from 1-36 are red; the other half black. Both 0 and 00 are green.

Roulette Suppose that a player places a simple $1 bet on red. If the ball lands in a red slot, the player gets the original dollar back, plus an additional dollar for winning.

Roulette Let’s define the random variable X= net gain from a single $1 bet on red. Possible values? Probability model? What is the player’s average gain? x -$1 $1 P(x) 20/38 18/38 Ordinary average of the two possible outcomes -$1 and $1 is $0. But $0 isn’t the average winnings because the player is less likely to win $1 than to lose $1. In the long run, the player gains a dollar 18 times in every 38 games. So long run average gain is: Mx= (0$1)(20/38)+($1)(18/38)= -0.05 You can see that in the long run, the player loses five cents per bet. (we call this the expected value)

Example: # of Courses for which a randomly chosen UA student is registered 1 2 3 4 5 6 7 P(Y) 0.02 0.03 0.09 0.25 0.40 0.16 0.05

𝜇 𝑋 = 𝑥 𝑖 𝑝( 𝑥 𝑖 ) 𝜎 𝑋 2 = 𝑥 𝑖 − 𝜇 𝑋 2 𝑝( 𝑥 𝑖 )

X 1 2 3 4 5 6 7 8 9 10 p(X) 0.001 0.006 0.007 0.008 0.012 0.020 0.038 0.099 0.319 0.437 0.053

Apgar score measures muscle tone, skin color, respiratory effort, strength of heartbeat, and reflexes 1 minute after a baby’s birth and again 5 minutes after. The scores range from zero to 10. The distribution of scores is listed in the table. What is the mean and stdev of this distribution? What is the probability that a child will be born within one stdev of the mean? More than 2 stdev above the mean? What percent of babies are born within one stdev? Katherine- is in distress, her cord is wrapped around her neck, she’s holding it and squeezing it, very low apgar score (6). Then do it again, apgar is better—she’s recovered Thomas- big red faced, peed all over my mom– apgar is 8 or 9 Elizabeth- apgar is lower than Thomas’, one thing they measure is how much you cry, and I didn’t cry at all (we were all within one stdev of the mean)