# Special Distributions Two Discrete Distributions Binomial and Geometric

Special Distributions Two Discrete Distributions: Binomial and Geometric One Continuous Distribution: Normal Distributions

Calculator needed!

Binomial Probability Distribution (BPD)

Suppose we decide to record the gender of the next 25 newborns at a particular hospital. t a h t e c an ? h e c l a e h m t e f s i e t r a a What Wh ast 15 is the e chance betwe at l that en 10 a nd 15 a re fem , s n ale? r o b ew pect n 25 e ex e h w t n f a o c t These questions can be y u n O ? a m male answered using a binomial w ho e fe distribution. b o t

Properties of a Binomial Experiment 1. There a fixed number of trials 2. Each trial results in one of two mutually We use n to denote the fixed number exclusive outcomes. (success/failure) of trials. 3. Outcomes of different trials are independent 4. The probability that a trial results in success is the same for all trials The binomial random variable x is defined as x = the number of successes observed when a binomial experiment is performed

FITS F------ Fixed I-------Independent T------Two Mutually exclusive S-------same Probability

Thumbs up for binomial distributions! 1) Toss a coin 10 times and count the number of heads Yes 2) Deal 10 cards from a shuffled deck and count the number of red cards No, probability does not remain constant 3) The number of tickets sold to children under 12 at a movie theater in a one hour period No, no fixed number

4. A certain surgical procedure has an 85% chance of success. A doctor performs the procedure on eight patients. The random variable represents the number of successful surgeries. Yes

5. A jar contains five red marbles, nine blue marbles and six green marbles. You randomly select three marbles from the jar, without replacement. The random variable represents the number of red marbles. No, NOT Independent.

Binomial Probability Formula: Let n = number of independent trials in a binomial experiment p = constant probability that any trial results in a success Where: Technology, such as calculators and statistical software, will also perform this calculation.

TI-nspire? 1. Menu 2. Probability 3. Factorial ! (5) (1) 4. Combination (3)

Example: In how many ways can we choose a team of 3 people from a group of 5? ANS: - Menu - Probability(5) - combination(3) - n. Cr(5, 3) - enter

Example-1: A coin is tossed 7 times. Find the probability of getting exactly 3 heads.

Instead of recording the gender of the next 25 newborns at a particular hospital, let’s record the gender of the next 5 newborns at this hospital. Is this a binomial experiment? Yes, if. What the births were notofmultiple births is the probability “success”? (twins, etc). Define the random variable of interest. x = the number of females born out of the next What will the largest of the Will a binomial random variablevalue always include thebinomial value of 0? 5 births random value be? What are the possible values of x? x 0 1 2 3 4 5

Newborns Continued. . . What is the probability that exactly 2 girls will be born out of the next 5 births? What is the probability that less than 2 girls will be born out of the next 5 births?

Newborns Continued. . . Let’s construct the discrete probability distribution table for this binomial random variable: x p(x) 0 Notice that 1 this is 2 the same 3 as multiplying 4 5 n×p. 03125. 15625. 03125 What is the mean number of girls born in the next five births? this is a discrete mx = 0(. 03125)Since + 1(. 15625) + 2(. 3125) + distribution, we could use: 3(. 3125) + 4(. 15625) + 5(. 03125) =2. 5

Population Parameters of a Binomial Distribution Mean: = np Variance: 2 = npq Standard Deviation: =

Newborns Continued. . . How many girls would you expect in the next five births at a particular hospital? What is the standard deviation of the number of girls born in the next five births?

Try Me: A survey indicates that 65% of Chinese women consider dancing as their favorite leisure time activity. You randomly select four women and ask them if dancing is their favorite leisure-time activity. Find the probability that (i) exactly two of them respond yes, (ii) fewer than two of them respond yes, (iii) at least two of them respond yes.

How to use GDC?

1. 2. 3. 4. TI-nspire Menu Probability (5) Distributions (5) Binomial pdf ………(to find prob of exactly one value) Or 5. Binomial cdf ----(to find multiple binomial prob like at most, at least problems )

1. Menu 2. Probability (5) 3. Distributions (5) 4. Binomial Pdf (D) Binomial pdf

1. Menu 2. Probability (5) 3. Distributions (5) 4. Binomial Cdf (E) Binomial cdf

Check with your GDC! A survey indicates that 65% of Chinese women consider dancing as their favorite leisure time activity. You randomly select four women and ask them if dancing is their favorite leisure-time activity. Find the probability that (i) exactly two of them respond yes, (ii) fewer than two of them respond yes, (iii) at least two of them respond yes.

The proportion of vegetarians in a big city is 0. 2. A random sample of 15 persons was selected. What is the probability that the sample contains a) 5 vegetarians? b) No vegetarian? c) At most 3 vegetarians?

To get a full credit on AP-exam, the following things should be identified: 1. Name ----Binomial (words or formula) Binomial, n= , p= 2. Parameters 3. Boundary 4. Direction P(x>3) or probability of more than 3 (3 is the boundary, more refers the direction)

Quick survey The pace of the lecture for this slide set was… a) Fast b) About right c) A little slow d) Too slow 27

Try Me A Stats 10 test has 5 multiple choice questions with four choices with one correct answer each. If we just randomly guess on each of the 5 questions, what is the probability that you get exactly 2 questions correct? a) 0. 6250 b) 0. 25 c) 0. 0625 d) 0. 2636

Why not me? The American Red Cross says that about 11% of the U. S. population has Type B blood. A blood drive is being held at your school. What is the probability that at least 2 of the first 10 blood donors has Type B blood? A) 0. 088 B) 0. 697 C) 0. 214 D) 0. 303

Binomial Probability Distribution • N Contains n trials(fixed number) • O Only Two outcomes: success or failure • P The probability of each trial is the same • I The trials are independent. • The binomial random variable is denoted as X

How to use GDC? TI-89 1. Math (shift 5) 2. Prob (7) 3. Choose factorial (1) or combination (3)

TI-89 1. Go to Stat/List editor 2. Press F 5 3. Scroll down and Select either (B) Binomial pdf ………(to find prob of exactly one value) Or ( C) Binomialcdf ----(to find multiple binomial prob like at most, at least problems )

TI-89 1. Go to Stat/List editor 2. Press F 5 3. Scroll down and Select either (B) Binomial pdf ………(to find prob of exactly one value) Or ( C) Binomialcdf ----(to find multiple binomial prob like at most, at least problems )

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