Binomial Probability Distributions

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Presentation transcript:

Binomial Probability Distributions

Def: Binomial probability distribution 1 Def: Binomial probability distribution 1. procedure has fixed # of trials 2. trials must be independent 3. each trial must have all outcomes classified into 2 categories 4. probabilities must remain constant for each trial

1. Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not binomial, state why (Similar to p.344 #7-16) Twenty students are asked how many tacos they eat each month.

2. Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not binomial, state why (Similar to p.344 #7-16) Fifty people are asked on whether they believe in reincarnation.

Notation: - P(S) = p (p=probability of a success in one trial) - P(F) = 1-p = q (q=probability of a failure in one trial) - n= # of trials - x = # of successes in n trials - P(x) = probability of getting exactly x successes among the n trials

Binomial probability formula (for x = 0, 1, 2, …., n):

3. Finding Probabilities (By Formula) n = 10, p=0.3, find the P(x = 4)

4. Finding Probabilities (By Formula) n = 10, p=0.2, find the P(x < 3)

5. Finding Probabilities (By Table) n = 4, p=0.3, find the P(x = 1)

6. Finding Probabilities (By Table) n = 4, p=0.45, find the P(x < 2)

7. Finding Probabilities (By Table) n = 4, p=0.45, find the P(x < 2)

TI-83/84: binompdf (2nd button – vars button) : of this form: binompdf(n,p,x) where n=# of trials, p=probability of success of one of the trials, and x is the number of successes you want to find) binomcdf (2nd button – vars button) : of this form: binomcdf(n,p,x) *same notation as above this adds all the probabilities from 0 to x (cumulative)

Finding Probabilities Identify number of trials (n) Identify probability of single trial (p). This is often given as a percentage List the x values the problem entails and rewrite with inequality/equality symbols Use the table on the following slide to determine how to enter the function

Forms Form TI-83/84 Function P(x = a) binompdf(n,p,a) P(x < a) binomcdf(n,p,a) P(x > a) 1-binomcdf(n,p,a-1) P(a < x < b) binomcdf(n,p,b)-binomcdf(n,p,a-1)

8. Examples Given a binomial distribution with n = 30 and p=0.25, find the following: P(exactly 8) P(less than 17) P(at most 12) P(more than 20 P(at least 25) P(between 10 and 20, inclusive)

BINOMIAL DISTRIBUTIONS

9. Find the mean, standard deviation and variance 30% of Cowley Students believe in ghosts. Compute the mean, standard deviation, and variance of the random variable X, the number of Cowley students that believe in ghosts based on a random sample of 100 students.