Section 5.2 Random Variables

Is life insurance worth the cost? Why Is This Important??? There is a 0.9986 probability that a randomly selected 30 year-old male lives through the year (based on data from the U.S Department of Health and Human Services). A Fidelity life insurance company charges $161 for insuring that the male will live through the year. If the male does not survive the year, the policy pays out $100,000 as a death benefit. Is life insurance worth the cost?

Game winning strategies Why Is This Important??? The television game show Deal or No Deal begins with individual suitcases containing the amounts $0.01, $1, $5, $10, $25, $50, $75, $100, $200, $300, $400, $500, $750, $1,000, $5,000, $10,000, $25,000, $50,000, $75,000, $100,000, $200,000, $300,000, $400,000, $500,000, $750,000, and $1,000,000. If a player adopts the startegy of choosing the option of “no deal” until one suitcase remains, the payoff is one of the amounts and they are all equally likely. What is the expected value for this strategy? Game winning strategies

Let’s Get Started … The Basics A random variable is a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure. A probability distribution is a description that gives the probability for each value of the random variable. It is often expressed in the format of a graph, table, or formula.

(Number of peas with green pods) For Example … The Secret Garden A Probability Distribution Mr. Llorens has a strange fascination with pea pods. Every summer, Mr. Llorens grows peas in his garden and keeps track of how many green pea pods and yellow pea pods each plant produces. Here is a sample of some of the results in his creeper journal. x (Number of peas with green pods) P(x) 0.01 1 0.015 2 0.088 3 0.264 4 0.396 5 0.237

Characteristics of a Probability Distribution The sum of all probabilities must be 1, but values such as 0.999 or 1.001 are acceptable because they result from rounding errors. Each probability value must be between 0 and 1 inclusive.

CHECK YO SELF!!! x P(x) 0.250 1 0.382 2 0.119 3 0.481 4 0.260 5 0.127

Section 5.2 only deals with discrete random variables. CAUTION!!! Section 5.2 only deals with discrete random variables.

REFRESH YO SELF!!! Examples??? A discrete random variable has either a finite number of values or a countable number of values, where “countable” refers to the fact that there might be infinitely many values, but they can be associated with a counting process so that the number of values is 0, 1 or 2 or 3, etc. A continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions. Examples???

Lightning Practice Round The number of tennis balls Brendan misses every time he goes to the tennis courts after school. The number of awesome hairstyles Tori can rock. The amount of coffee (in ounces) Sarah can drink every morning. The number of times Ryan says his name is Chris, not Ryan. The amount of time Casey spends googling T. Swizzle’s supposed relationship with One Direction band members. The amount of money Olivia spends on her badminton swagg.

Is life insurance worth the cost? REMEMBER THIS??? There is a 0.9986 probability that a randomly selected 30 year-old male lives through the year (based on data from the U.S Department of Health and Human Services). A Fidelity life insurance company charges $161 for insuring that the male will live through the year. If the male does not survive the year, the policy pays out $100,000 as a death benefit. Is life insurance worth the cost?

“Freaking Finally, Ms. P!!!” – Direct Quote by You Now that you have the background info … on to the MAIN EVENT Formula 5-1 Mean for a probability distribution Formula 5-2 Variance for a probability distribution (easier to understand) Formula 5-3 Variance for a probability distribution (easier to compute) Formula 5-4 Standard deviation for a probability distribution

So, How Do We Use These? Let’s Calculate! Remember Mr. Llorens and his creepy plant obsession? Let’s Calculate! x (Number of peas with green pods) P(x) 0.01 1 0.015 2 0.088 3 0.264 4 0.396 5 0.237 Find the mean, variance, and standard deviation for the probability distribution. What does this mean if these results are the number of green pods out of 5 possible pods?

IMPROVE YO SELF!!! Pg. 214-215 #5, 6, 7-10

Section 5.2 Random Variables

Calculator Mad Dog Skillz TI-83 / TI-84 Computing Mean and Standard Deviation Enter the values of the random variable x in the list L1. Enter the corresponding probabilities in the list L2. Press STAT, select CALC option and choose 1-Var Stats Enter "L1,L2" and Press the ENTER key.

Building Onto Previous Knowledge Important Definition The expected value of a discrete random variable is denoted by E, and it represents the mean value of the outcomes. It is obtained by finding the value of Where have we seen this before?

A Little (LOT) Clarification We can think of the mean as the expected value in the sense that it is the average value that we would expect to get if the trials could continue indefinitely. For example, an expected value of 3.2 girls is not meant to be interpreted as the number of girls in one trial will be equal to 3.2, but rather it means that among many such trials, the mean number of girls is 3.2

So how do we know what to expect? The Exciting Stuff So how do we know what to expect? Range Rule of Thumb Maximum usual value = Minimum usual value =

So how do we know what to expect? The Exciting Stuff So how do we know what to expect? Rare Event Rule x successes among n trials is an unusually high result when P(x or more) ≤ 0.05 x successes among n trials is an unusually low results when P(x or fewer) ≤ 0.05

Now Apply It! Based on data from CarMax. Com when a car is randomly selected the number of bumper stickers and the corresponding probabilities are 0 (0.824), 1(0.083), 2 (0.039), 3 (0.014), 4 (0.012), 5 (0.008), 6 (0.008), 7 (0.004), 8 (0.004), and 9 (0.004). Does the given information describe a probability distribution? Is it unusual for a car to have more than one bumper sticker?

Now Apply It! In the Illinois Pick 3 lottery game, you pay $0.50 to select a sequence of three digits, such as 233. If you select the same sequence of three digits that are drawn, you win and collect $250. How many different selections are possible? What is the probability of winning? Find your expected value of winning.

Now Apply It! There is a 0.9968 probability that a randomly selected 50- year-old female lives through the year (based on data from the U.S Department of Health and Human Services). A Fidelity life insurance company charges $226 for insuring that the female will live through the year. If she does not survive the year, the policy pays out $50,000 as a death benefit. If she purchases this policy, what is her expected value? Can the insurance company expect to make a profit from many such policies?

IMPROVE YO SELF!!! Pg. 216-217 #16, 17, 18, 28, 29