# Stat 200

Objectives: To become familiar with the application of probabilities and binomial probability distributions.

Introduction: Since probabilities were first developed for gambling purposes, we will consider a couple probability and odds problems first. Then we will look at look at binomial probability distribution and how that can help us answer questions about a population.

Instruction: Answer the following questions showing your work (calculation) since some of the answers can be found online.

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1. The LOTTO lottery used 69 white balls from which you chose five and then one power ball from which you choose one. Any given number for a white ball can only be chosen once. Hint: For A, look for the probability for each ball selection, then use the fundamental counting rule. For B think about the relationship between probability and odds.

A. What is the probability of a single ticket winning?

B. What are the odds of a single ticket winning?

C: If the prize in a given week is \$100,000,000 and a ticket cost \$3, what is the lotteries profit on just the grand prize not including smaller prizes?

2. Many people betting roulette at a casino have tried the doubling their losses strategy to get back to winning. Is this a good strategy? An American roulette wheel has the numbers 1 through 36 (half red and half black) plus 0 and 00 (green) for a total of 38 spaces or odds of 1 to 37, yet pays 35 to 1 giving the house ad 5.26% advantage. Playing black or red pays 1:1.

A. Many people betting roulette at a casino have tried the doubling their losses strategy to get back to winning. Is this a good strategy? If you start with \$540 and bet just red or black and you start with a \$10 bet, what is the probability you will lose all your money on consecutive bets and how many bets is that? Betting example: Lose \$10 on the first round then bet \$20 on the next round. Lose the \$20 and bet \$60 on the third round to cover the \$10 from the first round and the \$20 from the second round.

B. If you played 38 hands that overall went exactly with the odds, how much would you win or lose playing \$10 a hand.

C. If you play 38 rounds betting \$10 per round on just a single number, and the odds play out perfectly, how much will you win or lose?

D. What are the probability and odds of losing 38 consecutive rounds?

3. Create your own binomial distribution experiment. Go on the internet and find out about the distribution of some characteristic of a population. You could look at things like eye color, heights of males or females, color of m&ms. Then select your sample such as the people you work with, the people in your dorm, or how many m&ms you grab from a jar. You don’t have to collect any data. You just need to know the number of your sample or trial. Create a table for the probability distribution for each value. For example if you want to find out the probability of getting a certain number of blue m&ms in a handful of 20. Create a table of x= 0 through 20 and find the probability for each value of x. If your sample size is more than 30, just show the first 30 probabilities, but be sure to be clear how many trials you are using. Copy and paste your table below. Use at least a sample (trial) of at least 10. Then answer the following questions.

A. Describe your population and the probabilities for the characteristics of that population you are interested in, what variable you will be answering questions about, and how many trials (sample size).

B. What is the probability of getting none of your variable?

C. What is the probability of getting from 2 to 5 of your variable?

D. What is the probability of getting eight or more of your variable?

E. What is the probability of getting at least 2 of your variable?

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