Day 2.notebook November 25, Warm Up Are the following probability distributions? If not, explain.
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1 Warm Up Are the following probability distributions? If not, explain. ANSWERS Complete the probability distribution. Hint: Remember what all P(x) add up to? 4. Find the mean and standard deviation. x P(x) Determine whether the random variable is discrete (counted) or continuous (measured). a. The number of pencils in a pencil box. b. The amount of time you sleep through English class. c. The maximum speed of your 4 wheeler. d. The number of full moons in a year's time. e. The number of buttons on a shirt. Warm Up ANSWERS Are the following probability distributions? If not, explain yes 3. Complete the probability distribution. 1 ( ) = Find the mean and standard deviation. x P(x) μ = 2 σ = 1.2 no 5. Determine whether the random variable is Discrete (counted) or Continuous (measure). a. The number of pencils in a pencil box. D b. The amount of time you sleep through English class. C c. The maximum speed of your 4 wheeler. C d. The number of full moons in a year's time. D e. The number of buttons on a shirt. D
2 Expected Value: What you expect to get over time; the theoretical average of a probability distribution E(x) or μ Expected Value (mean) = Standard Deviation: Spread of the data in the probability distribution The probability that 0, 1, 2, 3, or 4 people will be placed on hold when they call a radio station is given in the table below. How many callers does the station expect to have on hold at one time?
3 More Expected Value One thousand tickets are sold at $1 each for a 42 inch TV valued at $350. What is the expected value of gain if a person purchases one ticket? WIN LOSE GAIN, x P(x) E(x) = More Expected Value A company is a defendant in a product liability case and must choose between settling out of court for a loss of $150,000 and going to trial. If they go to trial, they can pay out nothing (if found not guilty) or pay out $500,000 (if found guilty). The attorney estimates that the probability of a not guilty verdict is 0.8. Suppose the defendant goes to trial, find the expected value of the amount lost. GAIN ($), x P(x) E(x) = Not Guilty Guilty
4 Probabilities that involve Casino Games Games in gambling casinos have probabilities and payoffs that result in negative expected values. This means that players will lose money in the long run. With no clocks on the walls or no windows to look out of, casinos bet that players will forget about time and play longer. This brings casino owners closer to what they can expect to earn, while negative expected values sneak up on losing patrons (they do provide free beverages). Population of Las Vegas: 535,000 Slot Machines in Las Vegas: 150,000 Avg Earnings per Slot Machine per Year: $100,000 Avg Amt Lost per Hour in Las Vegas Casinos: $696,000 Avg Amt an Adult American Loses Gambling per Year: $350 Avg Amt an Adult Nevada Resident Loses Gambling per Year: $1000 Probabilities that involve Casino Games KENO: house wins $0.27 for every $1 bet CHUCK A LUCK: house wins $0.52 for every $1 bet CRAPS: house wins $0.88 for every $1 bet ROULETTE: house wins $0.92 for every $1 bet "HOUSE" = casino "PLAYER" = average Joe A "fair" game would mean the expected value is. If the expected value is positive, the wins most of the time. If the expected value is negative, the wins most of the time.
5 More Expected Value You travel to Las Vegas to hit the casinos. Instead, you get hooked up with the mob and play a numbers game. You place a bet on a 3 digit number of your choice. The typical payoff is 499 to 1, meaning that for every $1 bet, you would get $500 (your net return is $499). Suppose you bet on the number 327. What is your expected value of gain/loss? WIN LOSE GAIN ($), x P(x) E(x) = 1. The expected value of a probability distribution is What are the 2 requirements for a probability distribution? a. b. 3. Find the mean and standard deviation for the probability distribution below.
6 4. Student council sold tickets for $4 each to raise money for the Prom. They sold 750 tickets. The prize for the winning ticket is $200. Find the expected value of gain for the person who buys a ticket. How much money will the student council raise? WINS LOSES Gain ($), X P(X) 5. If a 60 year old buys a life insurance policy worth $1000 at a cost o $60 and has a probability of of living to age 61, find the expectation of the policy. LIVES DIES Gain ($), X P(X)
7 6. A local fire department wants to raise $2000 for new equipment. They sell raffle tickets for $2 each for a cash prize of $1000. If 1200 tickets are sold, find the expected value. Are they selling enough ticke to make their goal of $2000? WINS LOSES Gain ($), X P(X)
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