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Chapter 17

Chapter 17. Probability Models The Geometric Distribution.

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Chapter 17

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  1. Chapter 17 Probability Models The Geometric Distribution

  2. Suppose there is a particular toy that your neighbor wants to get from a McDonald’s Happy Meal. Assume that the toys are randomly distributed and the probability of getting the toy is 15%. How many Happy Meals do you need to buy in order to get the toy? This situation is considered a Bernoulli trial and all Bernoulli trials have the following three characteristics: There are only two outcomes (success and failure). Either he gets the toy or he doesn’t. The probability of success, denoted p, is the same on every trail. In other words, the probability doesn’t change. The trails are independent. Getting a different toy from the first box does not affect what will happen on the next box. You must ALWAYS check these three assumptions when dealing with these types of problems! Introduction

  3. Here are a few examples of Bernoulli trials: We use a coin to see which of two football teams gets the choice of kicking off or receiving to begin a game. A young couple prepares for their first child; the possible outcomes are {boy; girl}. Make sure that you always check your assumptions and conditions! There are two types of Bernoulli trials that we will look at in this chapter: binomial and geometric distributions. These discrete random distribution will be distinguished by the way you define your random variable. Introduction

  4. A geometric random variable is a variable whose domain contains the number of trials that are needed to achieve the first success in a repeated trial experiment. Geometric Distributions

  5. In order to calculate probabilities using the Geometric distribution, we must satisfy the same three conditions for Bernoulli Trials. The whole point of the geometric model is to look at situations where we are dealing with “first successes.” Many times we wish to determine the average number n of trials required to obtain the first success. This is the Expected Value (or the mean) of the first success. Other times, we want to know the probability of getting our first success in n number of trials. Either way, we still are dealing with “first successes.” Geometric Settings

  6. Let X be a geometric random variable with probability of success p on each trial. The mean µ, or expected value, of the random variable is (this is the expected number of trials required to get the first success) The variance of X is Standard deviation is Geometric Random Variable:Mean and Standard Deviations

  7. Rule for Calculating Geometric Probabilities In order to calculate the probability of the first success in n trials, we use the following: P(X = n) = (1 - p)n-1p or P(X = n) = qn-1p Example: If you roll a die, what is the probability that the first 6 will be on the fourth roll? We can also use the calculator: geometpdf (p, n) geometpdf (1/6, 4) ≈ 0.0965

  8. Independence • One of the most important requirements Bernoulli Trials is that each trial must be independent. • Most times in probability, we can assume that events are independent (such as rolling two dice). • However, in real-life Statistics, independence is compromised since we choose our sample without replacement. This means that once you choose an individual for a study, the probability for choosing the next individual changes. • So what do we do?

  9. Independence • Well, as long as our population is large enough, removing a few individual is negligible. 1 out of a billion and one is not much different than 1 out of a billion. • OK, but how big is large enough? There is a condition or a criteria that we can use to determine if a population is large enough: • The 10% condition: If independence is violated, it is still OK to utilize Bernoulli Trials as long as the sample is smaller than 10% of the population. • As long as our sample is not too big, the probabilities don’t change significantly.

  10. More Probability Rules • The probability that it takes more than n trials to see the first success is P(X > n) = (1 - p)n • Example: • Let’s say you pick a card with replacement from a standard 52-card deck. What is the probability it will take more than 10 tries before you get your first heart? • There is approximately a 5.6% chance that it will take more than 10 tries before we see our first heart.

  11. Only 6% of people have type O-negative blood. If donors line up for a blood drive, how many people should we expect to see before we see our first O-negative blood? What is the probability that it will take more than 10 people before we see the first O-negative donor? What is the probability that a type O-negative donor is found within the first four people in line? What is the standard deviation? Example:

  12. First things first, ALWAYS…. Check your Assumptions (and conditions): There are two outcomes: Success = A person has type O-negative blood Failure = A person has different type blood The probability for success is the same for each person Each person has a .06 chance of success The trails are independent Actually, the trails are not independent!!! Since n is less than 10% of the population (we don’t plan on drawing blood from more than 10% of the population!), this conditionmaybe compromised, so we’re ok. Solution

  13. Solution • Only 6% of people have type O-negative blood. If donors line up for a blood drive, how many people should we expect to see before we see our first O-negative blood? What is the probability that it will take more than 10 people before we see the first O-negative donor? What is the probability that a type O-negative donor is found within the first four people in line? What is the standard deviation?

  14. Solution • Only 6% of people have type O-negative blood. If donors line up for a blood drive, how many people should we expect to see before we see our first O-negative blood? What is the probability that it will take more than 10 people before we see the first O-negative donor? What is the probability that a type O-negative donor is found within the first four people in line? What is the standard deviation? Or you can use 1 – geometcdf(.06,10)

  15. Solution • Only 6% of people have type O-negative blood. If donors line up for a blood drive, how many people should we expect to see before we see our first O-negative blood? What is the probability that it will take more than 10 people before we see the first O-negative donor? What is the probability that a type O-negative donor is found within the first four people in line? What is the standard deviation? Or you can use geometcdf(.06, 4)

  16. Solution • Only 6% of people have type O-negative blood. If donors line up for a blood drive, how many people should we expect to see before we see our first O-negative blood? What is the probability that it will take more than 10 people before we see the first O-negative donor? What is the probability that a type O-negative donor is found within the first four people in line? What is the standard deviation?

  17. Assignment

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