3. Definitions A Binomial Setting is an scenario where each of the following are true Each observation is either a ‘success’ or a failure’ (2 outcomes) There are a fixed number of observations (n) Observations are independent The probability of success is the same for each observation (p)
4. Definitions The Binomial Distribution X = # of successes from a binomial setting Abbreviated with “B(n, p)”where n = number of trails, p = prob. of success
5. Definitions The Binomial Distribution Examples “X = The number of heads when 5 coins are flipped”B(5, 0.5) “X = The number of heart cards drawn from a standard deck with replacement after 3 tries.”B(3, ¼) “X = The number of working computer chips in a set of 8 chips if the manufacturer has 0.1% defects”B(8, 0.999)
6. Sampling Distribution of a Count Suppose an SRS of size n is drawn from a population with a proportion p for success. Unless there is replacement, this isn’t binomial. Why? If the population is much greater than the sample size, then the sample has an approximate binomial distribution B(n, p) This helps use the binomial distribution in many cases that aren’t exactly a binomial setting.
7. Formulas for Binomial Settings Using a tree diagram, it is possible to formulate the binomial distribution. Example X = number of heads when a loaded coin is flipped 3 times (P(heads = 0.3)
8. Formulas for Binomial Settings Setting is B(3, .3) Sample Space- HHH, HHT, HTH, THH, HTT, THT, TTH, TTT ProbabilitiesP(X = 3) = 0.027, P(X = 2) = 3·(.3)2(.7) = 0.189, P(X = 1) = 3·(.3)(.7)2=0.441, P(X = 0) = (.7)3=0.343 Unfortunately, this works best only for simple settings
9. Formulas for Binomial Settings Binomial coefficient This is a part of the formula for the binomial distribution. n = # trails, k = # of success You may have seen this formula before!
10. Formulas for Binomial Settings Binomial coefficient Your calculator can find this coefficient for you [math] , “PRB,” “nCr” example
11. Formulas for Binomial Settings Binomial Probability If we have a binomial setting B(n, p) and we want to know P(X = k)
12. Formulas for Binomial Settings Binomial Probability Actually your calculator is very efficient in these caculations.The binomial magic is [2nd] [vars] (dist), “binompdf(“ Again assuming B(n. p)
13. Some Alphabet Soup “pdf” means “probability distribution function,” which is exactly what we are doing! “binompdf” is the “binomial probability function” This is a discrete probability distribution “cdf” means “cumulative distribution function” This will add together a number of successes.
14. Cumulative Binomial Distribution For the question, “if B(n, p), what is the probability k or less successes?” B(5, 0.33) P(X < 2) = P(X=0) + P(X=1) + P(X=2) = binomcdf(5, 0.33, 2) Pay attention to where the “equals” goes! Binomcdf(n,p, k) is always P(X < k)
15. Cumulative Binomial Distribution If B(6, 0.25), what is the probability of less than 4 successes? B(6, 0.25) P(X < 4) = P(X<3) (always rewrite as “<“) = binomcdf(6, 0.25, 3) =0.9624 Pay attention to where the “equals” goes! binomcdf(n, p, k) is always P(X < k)
16. Cumulative Binomial Distribution For cases involving ‘X>k,’ or ‘X>k’ use the property of complimentary sets Pay attention to where the “=” goes! In B(100, 0.95), what is the probability of more than 90 successes? B(100, 0.95) P(X > 90)= 1- P(X<90) = 1 - binomcdf(100, 0.95, 90) =0.9718
17. Mean and Standard Deviation For a binomial distribution B(n, p) the following formulas hold: Remember that these are only for a binomial distribution We should also note that can be thought of as the “expected value”
18. Normal Approximation You should have noticed by now that the Binomial distribution produces a single peak distribution If p is within a certain set of numbers, the distribution is relatively symmetric. Because we like to use the Normal distribution, we have conditions under which the binomial distribution is approximately Normal
19. Normal Approximation A binomial distribution is approximately Normal N(np, (npq)) when both np>10 and nq> 10. When using the Normal dist to approximate, be sure to: state “Distribution is approximately Normal: N(np, (npq))” Show that the two conditions above are met Remember that this is an approximation, but it is most often good enough
21. The Geometric Setting The geometric setting is almost like the binomial setting with one major difference: Instead of asking “how many successes,” we ask, “when is the first success?”
22. The Geometric Setting Observations are either “success” or “failure” The observations are independent The probability of success is the same for each observation The variable of interest is the number of trails until the first success
23. Geometric Distribution If a random variable X satisfies the geometric setting, then we call the distribution of X a geometric distribution FormulaP(X = k) = q(k-1)p notice that this is (k-1) failures and one success
24. Geometric Distribution on the TI Like the Binomial Distribution, the Geometric Distribution is found at[2nd] [var] (dist) for G(p)P(X=k) = q(k-1)p = geompdf(p,k)
25. Geometric cdf The probability that the first success is within the first k trails can be given with: G(p) P(X < k) = P(X=0) + P(X=1)+ … +P(X=k) =geomcdf(p, k)
26. Geometric cdf What is the probability that the first “six” is rolled before four throws of a die? Pay attention to the “=“ sign G(1/6) P(X < 4) = P(X < 3) = geomcdf(1/6, 3) = 0.4213
27. Geometric cdf Use the compliment properties to find P(X>k) G(p) P(X > k) = 1 – P(x < k) And… pay attention to the ‘equal’ sign!!
28. Geometric cdf What is the probability that we roll a number less than 3 after 5 throws? G(2/6) (this corresponds to ‘1’ and ‘2’) P(X > 5) = 1 – P(X < 5) = 1 – geomcdf(2/6, 5) = 0.1317
29. Geometric cdf Alternatively, the probability that it takes more than k trails to see the first success can be given by: P(X > k) = (1 – p)k
30. Mean and Standard Deviation Mean of a geometric distribution is given by: = 1/p This is the expected value for the first success “on average, the first success occurs on the 1/p trail”
31. Mean and Standard Deviation Standard Deviation: This is not a Normal distribution, so don’t try to calculate z-scores and Normalcdf!
