Using Bernoulli Trials Geometric and Binomial Probability Models

Bernoulli Trials Both the Geometric and Binomial distributions require a scenario that fits a “Bernoulli trial.” What are the three requirements? There are only two possible outcomes (success or failure) The probability of success, denoted p, is the same for each trial The trials are independent For every scenario you encounter, you must read into it and analyze whether or not it fits a “Bernoulli trial” situation.

Example Analyses Consider the question of whether trials are independent when we toss a coin repeatedly. We can’t think of any reason why each flip wouldn’t be independent of the others, so we assume the flips are independent. We can confirm the Large Sample Assumption with the Success/Failure Condition np > 10 and nq > 10 The Independence Assumption is violated whenever we sample without replacement, but is overridden by the 10% Condition. As long as we don’t drain off more than 10% of the population, the probabilities don’t change enough to matter.

AP Exam Hint On the AP Exam students are required to check, not just state, the conditions. It’s not sufficient to simply write np > 10 and put a check mark there. You must show you actually checked it using the values given in the question. Example: np = 220(0.75) = 165 > 10.

The Geometric Model X = the # of trials required until the first success occurs. Because “n” is not fixed, there could be an infinite number of X values. However, the probability that X is a very large number is more and more unlikely. Therefore, the probability histogram for a geometric distribution is always skewed right. Expected value: 𝑬 𝑿 = 𝟏 𝒑 Standard deviation: 𝑺𝑫 𝑿 = 𝟏−𝒑 𝒑 Probability that X is equal to x is given by the following formula: 𝑷 𝑿=𝒙 =(𝟏−𝒑 ) 𝒙−𝟏 𝒑

Geometric Calculator Tips Geometpdf(p,x) – allows you to find probability of any individual outcome on an exact trial. Geometcdf(p,x) – allows you to find probability of successes on or before the xth trial.

Example 1: Spam and the Geometric Model Postini is a global company specializing in communications security. The company monitors over 1 billion messages per day and recently reported that 91% of e-mails are spam. Overnight your inbox collects e-mail. When you first check your e-mail in the morning, about how many spam e-mails should you expect to have to wade through and discard before you find a real message?

Example 1: Spam and the Geometric Model Postini is a global company specializing in communications security. The company monitors over 1 billion messages per day and recently reported that 91% of e-mails are spam. What’s the probability that the 4th message in your inbox is the first one that is legitimate?

The Binomial Model X = the # of successes in “n” observations. Expected value: 𝑬 𝑿 =𝒏𝒑 Standard deviation: 𝑺𝑫 𝑿 = 𝒏𝒑𝒒 Probability of exactly “k” successes in “n” trials is: 𝑷 𝑿=𝒌 = 𝒏 𝒌 𝒑 𝒌 𝒒 𝒏−𝒌 where 𝒏 𝒌 = 𝒏! 𝒌! 𝒏−𝒌 !

Binomial Calculator Tips Binompdf(n,p,x) – allows you to find probability of any individual outcome. Binomcdf(n,p,x) – allows you to find the total probability of getting x or fewer successes among n trials.

Example 2: Spam and the Binomial Model Suppose your inbox contains 25 messages. What are the mean and standard deviation of real messages you should expect to see in your inbox?

Example 2: Spam and the Binomial Model Suppose your inbox contains 25 messages. What’s the probability that you’ll find only 1 or 2 real messages?

Example 3: M&Ms & Determining Your Model A new sales gimmick has 30% of the M&Ms covered with speckles. These special candies are mixed randomly with the normal candies as they are put into the bags for distribution and sale. You buy a bag and remove candies one at a time looking for the speckles. What’s the probability that the first speckled one is the fourth one we get? The tenth? What’s the probability that the first speckled one is one of the first three we look at? How many do we expect to check, on average, to find a speckled one?

Example 3: M&Ms & Determining Your Model What’s the probability that we’ll find two speckled ones in a handful of five candies? (These many different possible outcomes are called combinations.)