Ch. 16

Imagine selecting a US high school student at random. Define the random variable X=number of languages spoken by the randomly selected student. (a) make a histogram of the probability distribution. describe what you see. (b) what is the probability that a randomly selected student speaks at least 3 languages (c) compute the mean of the random variable X and interpret this value in context. (d) compute and interpret the standard deviation of the random variable X

(a) right skew, median at 1, languages vary from 1 to 5 (b) P(x≥3)=P(x=3) + P(x=4) + P(x=5) = 7.5% (c) E(x)=1.457 -if we randomly select many, many high schoolers we would expect average number of language spoken to be 1.457 (d) σx= 0.67 -the number of languages spoken by randomly selected high schooler typically varies about 0.67 away from the mean.

standard deviation of a discrete random variable

-since we use the mean as the measure of center for a discrete random variable, we use the standard deviation as our measure of spread. The definition of the variance is similar to the definition of the variance for a set of quantitative data -To get the standard deviation, take the square root of the variance (var(x)=(σ(x))²) -(don't have to divide by number because you are already multiplying by probability) var(x)=∑(xₙ - µx)²Pₙ σ(x)=√∑(xₙ - µx)²Pₙ

random variable

-takes numerical values that describe the outcomes of some chance process -the probability model/distribution of a random variable gives its possible values and their probabilities

ex of probability model

-tossing a fair coin 3 times -assign a variable to enumerate the outcome/that event -define x to the number of heads obtained (random variable is being described by a number instead of words) -so, X=0 instead of TTT X=1 instead of HIT THT TTH X=2 instead of HHT HTH THH X=3 instead of HHH -sample space is {x=0-4} -prob of having 0: 1/8 -prob of having 1: 3/8 -prob of having 2: 3/8 -prob of having 3: 1/8 -discrete because finite sample space because x can only be 0,1,2,4 -to be a legitimate probability model, have to add up to 1 (which it does 1/8 + 1/8 + 3/8 + 3/8 + 1/8) What is the probability that we get at least one head in three tosses of the coin? -we can use P(x≤1)= 1-P(x=0) instead of P(at least one H) = 1-P(none) -P(x≤1)= 1-P(x=0)= 1 - 1/8= 7/8

probability model graph

-value of x is on the x axis -probability is on y axis -if symmetric distribution/bell shaped, then middle of distribution (center) tells you the value of x that you can expect to see -we can use the mean as a measure of center to see what payout we would "expect" to see

mean (expected value) of a discrete random variable

-when analyzing discrete random variables, we follow the same strategy we used with quantitative data. Describe the shape, center, and spread, and identify any outliers. -the mean of any discrete random variable is an average of the possible outcomes, with each outcome weighed by its probability. -to find the mean (expected value) of x, multiply each possible value of x by its probability, then add all the products -multiply x to their respective probability and all them all up and get the mean (don't have to divide by number because you are already multiplying by probability) µx=E(x) = ∑xₙPₙ

The probability pₙ must satisfy two requirements:

1. every probability pₙ is a number between 0 and 1 2. The sum of the probabilities is 1 -for every probability model to be legitimate, the sum of all the probabilities must equal 1 or 100% -to find the probability of any event, add the probabilities pₙ ex: legitimate probability model P(X1 + X2 + X3 + ... Xn) = 1= 100% (this is your sample space)

probability model

Describes the possible outcomes of a chance process and the likelihood that those outcomes will occur -gives us all the possible outcomes of the events and the probability associated with them -what are all the values for x and what are all the assigned probabilities to each of those possible outcomes?

discrete random variable and their probability distributions

X takes a fixed set of possible values with gaps between. The probability distribution of a discrete random variable X lists the values xₙ and their probabilities pₙ -discrete if the sample space for x is finite (x can only be a limited number of values)

It costs $5 to play a game. You draw a card from a standard deck of cards. If you draw the Ace of Hearts, you win $100. All other aces win $10, and all other hearts win $5. Would you be willing to play? (a) what is the expected value (mean) and standard deviation of the payouts?

X=-5=costs $5 to play/draw any other card not listed; P(x)=36/52=69% X=100=win $100 from drawing A of H ; P(x)=1/52=2% X=10=win $10 from drawing all other aces ; P(x)=3/52=6% X=5=win $5 from drawing all other hearts; P(x)=12/52=23% -discrete model since x={-5,100,10,5} is finite -legitimate probability model because all the probabilities add up to 1 -graph is right skewed and the likely center is around $0 to $5 winnings so not likely going to win (a)Let's say 52 people (to represent 1 card in a deck) played this game and we want to average their winnings: µx = [(36)(-5) + (1)(100) + (3)(10) + (12)(5)]/52 = 0.19 Average winnings is about $0.19 (expected value). Standard deviation is how far winnings deviate from the average winnings σx= √([36(-5-0.19)^2 + (100-0.19)^2 + 3(10-0.19)^2 + 12(5-0.19)^2]/52) = 14.86 (which means you can expect to win 0.19±14.86, so you can lose around $13 or win $15) -all divided by 52 since there's 52 players

North Carolina State University posts the grade distribution for its courses online. 5 students in Statistics 101 in a recent semester received 26% A's, 42% B's, 20% C's, 10% D's and 2% F's. Choose a Statistics 101 student at random. The student's grade on a four point squat with A=4 is a discrete random variable X with this probability distribution. 1. say in words what the meaning of P(x≥3) is. What is the probability? 2. write the event "the student got a grade worse than C" in terms of values of the random variable X. What is the probability of this event?

X=0=F ; P(x)=0.02 X=1=D ; P(x)=0.1 X=2=C ; P(x)=0.2 X=3=B ; P(x)=0.42 X=4=A ; P(x)=0.26 -finite sample space (only 5 possibilities of x), so discrete model (doesn't keep going on forever) -legitimate probability model bc all the probabilities add up to 1 1. Probability that the student got at least a B/probability that the student gets either A or B. P(x≥3)=0.68 2 P(x<2)=P(0) + P(1) = 0.02 + 0.10 = 0.12

On Valentine's day, the Quiet Nook restaurant offers a Lucky Lovers Special that could save couples money on their romantic dinners. When the waiter brings the check, he'll also bring the four aces from a deck of cards. He'll shuffle them and lay them out face down on the table. The couple will then get to turn on card over. If it's a black ace, they'll owe the full amount, but if it's the ace of hearts, the waiter will give them a $20 Lucky Lovers discount. If they first turn over the ace of diamonds, they'll then get to turn over one of the remaining cards, earning a $10 discount for finding the ace of hearts this time. Based on a probability model for the size of the Lucky Lovers discounts the restaurants will award, what's expected discount for a couple? What's the standard deviation of the discounts?

x=how much of a discount you will get X=0=drawing a black ace or drawing an ace of diamonds then drawing anything other than an ace of hearts (earning no discount); P(x)=2/3 X=20=drawing an ace of hearts (earning $20 discount); P(x)=1/4 X=10=drawing an ace of diamonds then an ace of hearts (earning $10 discount) ; P(x)=1/12 -discrete model because x is finite (only 3 values that it can be) pick a card - 1/4 H ; 1/4 D (1/3 C, 1/3 S, 1/3 H (remaining cards after picked D)); 1/4 C; 1/4 C P(x=0)=P(AofC) + P(AofS) + P(AofD n AofC) + P(AofD n AofS) = 1/4 +1/4 + (1/3 x 1/4) + (1/3 x 1/4) = 8/12= 2/3 E(x)=0(2/3) + 20(1/4) + 10(1/12) = 5 + 10/12 = 5 5/6 = $5.83 (expect to win around $5.83) σ(x)=√2/3(0-5.83)² + 1/4(20-5.83)² + 1/12(10-5.83)² = $8.62 We can expect to save anywhere from $0 to $14