The Binomial Distribution

A motivating example… 35% of Canadian university students work more than 20 hours/week in jobs not related to their studies. This can have a serious impact on their grades. What is the probability that I have at least 5 such students in this class? Answer: There is better than a 99% chance!

What is a Binomial Distribution? Any random statistic that can be cast in a “yes/no” format where: N successive choices are independent “yes” has probability p and “no” has probability 1-p fits a binomial distribution. Suggest 3 other examples of data sets that can be modeled as binomial distributions

Looking a bit deeper… Suppose someone offered you the following “game”: Should you accept the bet? What is your expected return on this bet? How can we calculate the odds? Toss a coin 5 times. If you get 3 heads I pay you a dollar, otherwise you pay me 50 cents. I pay you a dollar, otherwise you pay me 50 cents.

Pascal to the rescue! There are exactly 10 ways to get 3 heads What is the probability of flipping 6 tails in 8 trials?

How to generate Pascal’s Triangle Pascal’s triangle “unlocks” the mystery of binomial distributions The cells in the triangle represent binomial coefficients which also represent all possible “yes/no” combinations In “math-speak” we use the following notation to calculate the number of ways “k” events can occur in “n” choices: Factorial notation 5! = 5x4x3x2x1 = 120 How many ways can 3 people be selected from a class of 39?

Math detail (FYI) The general binomial probability is: The Binomial Table is built from these terms Example: B(9,0.4),what is P(5)?

How to use the binomial distribution Assign “yes” and “no” and their respective probabilities to the instances in your problem Assign “n” and “k” and either use the formula, look up in a table or use a stats package (Excel works well) Example: ways: Look up in table Use formula Use Excel

From Binomial to Normal Distributions Binomial is a discrete probability distribution Normal is a continuous distribution When n becomes very large we can often approximate by using a N(  ) dist. How large is “large”? Rule of Thumb: when np >= 10 and n(1-p) >= 10 we can use the Normal Distribution approximation

Sample Proportions… We often are interested in knowing the proportion of a population that exhibits a specific property (statistic). We denote this the following way: p is a proportion (often interpreted as a probability) and is therefore a number between 0 and 1

Mean and Standard Deviation of a Sample Proportion If p is the proportion of “successes” in a large SRS of n samples, then: Look at Example 5.7

Working through some examples… 5.19: ESP A) ¼ = 0.25 B) p(10)+p(11)+…+p(20) or… 1- [p(0)+…p(9)], this can be read from Table C or done in EXCELEXCEL C) use You would expect 5 correct choices with a standard deviation of D) Since the subject knows that all 5 of the shapes are on the card the choices are no longer random and hence a binomial model is not appropriate – this was not the case in parts a-c

5.21 A) just use B) now use: C) D) p = 0.01  z = 2.33, use

Odds on the Oil! In order to make the play-offs, the Oilers must win 12 of their remaining 17 games. What is the probability that they will be successful? They currently have won 33 of the past 63 games. Step 1: re-word as a binomial distribution question, identify “n” and “k” Decide on what probabilities you will need to calculate Use either tables, Minitab or EXCELEXCEL

Odds on the Oil! – normal approximation Let’s look at using the normal approximation to solve this: In order to make the playoffs the oilers must have a better winning average than 33/60! However, at their current rate, how many of the 17 games do you expect them to win? What’s the standard deviation of this? Determine a z-score from this and comment on the likelihood of the Oiler’s success. Look at the sub-section “continuity correction” on pg 379 to help answer this. Should we expect this to give a reasonable answer?

5.24 Identify relevant statistics: n = 1500, p = 0.7 A) X = np = (1500)(0.70) = 1050 B) z = ( )/17.748,  better than 99% chance C) z = ( )/17.748,  NO CHANCE!!!!! D) X = np = 1190 and  = 18.89, chance that more than 1200 accept is now pretty good (p = )

In conlusion… Be sure that you understand what a binomial distribution is and when it can be applied Be able to use the probability equation on page 382 Know how to read and apply a binomial probability table (Appendix C) Know what Pascal’s triangle is and how it relates to binomial distributions Be able to relate the binomial distribtion to the normal distribution and when you can approximate with a normal distribution z-score analysis