Part IV: Randomness and Probability Chapter 16: Probability Models pages 413 – 418
AP Standard IIIA: Determine the probability of discrete random variables using geometric and binomial probability models.
I. Return Chapter 15 Quiz on Random Variables II. In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted, and each trial is independent. It is named after Jacob Bernoulli, a Swiss mathematician of the 17th century. Each observation falls into one of just two categories: o success: o failure: The probability of success is the same for each observation: o p Each trial is independent (or the population is at least ten times the sample size)
A. Examples 1. tossing a coin
2. looking for defective products rolling off an assembly line
3. shooting free throws in a basketball game
II. Suppose a cereal manufacturer puts pictures of famous athletes on cards in cereal boxes to increase sales. 20% of the boxes will feature LeBron James, 30% Danica Patrick and the rest, Serena Williams. How many boxes do you expect to have to open until you get a LeBron James card? A. In Chapter 10, we simulated the number of boxes we’d need in order to get a complete set of all three athletes’ cards. That’s pretty complex and well suited for simulation. However, the question here can be answered more directly by using the geometric model. A random variable X is geometric as long as all of the following four conditions are satisfied:
1. Each observation must fall into one of just two categories. In this scenario,
success: ______
failure: ______
2. The probability of success is the same for each observation. Here we have p ______
3. The variable of interest is the number of trials required to obtain the first success.
4. Each trial is independent
1. This is a reasonable assumption when tossing a coin or rolling a die or looking at situations like randomly selecting cards with replacement
2. However, it is still OK to procced if the assumption of independence is violated as long as the population is at least 10 times the sample size. We will need to make sure Rule of Thumb #1 N 10n is satisfied when randomly selecting samples without replacement. Read The 10% “Rule” on pages 414 – 415 B. Mean and Standard Deviation of a Geometric Random Variable
1 1 p 1. Formulas: E(X ) and p p 2
2. So using the formula we can answer the question. How many boxes do you expect to have to open until you get a LeBron James card?
C. Calculating Geometric Probabilities Formula: P ( X n ) ( 1 p ) n 1 p where n equals the number of trials it takes until the first success to occurs
1. We know the probability that you find a picture of LeBron in the first box of cereal is P(X 1) 0.2
2. What’s the probability that you don’t get LeBron until the second box you open?
3. What’s the probability that you don’t find LeBron until the fifth box?
III. “For Example” pages 415 – 416: Postini is a Global Company specializing in communications security. It reported that 91% of emails are spam.
A. First, check that the conditions required for Bernoulli trials
Each observation falls into one of just two categories where
Success is
Failure is
The probability of success is the same for each observation so p
We’ll consider each trial independent because
B. Overnight, your inbox collects email. When you first check your email in the morning, about how may spam emails should you expect to have to wade through and discard before you find a real message?
C. What’s the probability that the fourth message in your inbox is the first one that isn’t spam?
III. Read “Step-by-Step” Example page 417 to see another example
IV. Homework: #’s 1 – 2 and #’s 7 – 10 all pages 430 – 431