binomial day 1.notebook December 10, 2013 Probability Quick Review of Probability Distributions!

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Probability Binomial Distributions Day 1 Quick Review of Probability

# boys born in 4 births, x 0 1 2 3 4 Probability, P(x) 0.0625 0.

The sum of the probabilities of all the events in the sample space

1 Probability Binomial Distributions Day 1 Quick Review of Probability Distributions! # boys born in 4 births, x Probability, P(x) TWO REQUIREMENTS FOR A PROBABILITY DISTRIBUTION 1. The sum of the probabilities of all the events in the sample space must be equal to 1; that is: 2. The probability of each event in the sample space must be between or equal to 0 and 1; that is:

2 There is a special type of probability distribution: Binomial Distribution 1. Fixed number of trials (say, tossing a coin 20 times; n = 20) 2. Each trial is independent of the next one. 3. There are only two outcomes: a success (getting what you wanted to get) and a failure 4. The probability for a success is the same each time a trial is completed. Are the following BINOMIAL experiments? 1) Having 3 girls out of 5 births 2) Tossing a coin and recording the number of heads 3) Drawing a red marble out of a bag without replacement 7 red, 3 green, 4 blue marbles are in the bag 4) Guessing the correct answer on a multiple choice test An application of the binomial distribution: A new medical procedure is to increase a couple's chance of having a girl from 49% to 75%. What is the expected number of girls from 64 couples? Is it unusual to have 45 girls? What is the probability of having at least 45 girls out of 64 babies born?

Expected Value will be equal to the probability times the number of trials: E(x) = p n IN Calculator Mode: menu,

3 The Binomial Probability Formula n = # of times experiment is done x = # of outcomes you want p = probability of success q = probability of failure ( 1 p) *This formula is built in to the calculator. Expected Value will be equal to the probability times the number of trials: E(x) = p n IN Calculator Mode: menu, probability, distributions, binomial pdf This function will calculate the binomial probability of x successes from n trials. Exact Probability

IN Calculator Mode: menu, probability, distributions, binomial cdf This function will calculate the binomial probability of x or fewer successes from n trials.

4 IN Calculator Mode: menu, probability, distributions, binomial cdf This function will calculate the binomial probability of x or fewer successes from n trials. Cumulative Probability Example: A coin is tossed 2 times. Find the probability of getting 2 heads out of the 2 tosses. n = # heads, x Probability P(x) BY HAND: p = x = not difficult since we are tossing a coin only 2 times... Sample Space: S = {HH, HT, TH, TT}

By hand, this would be tedious since there are 64 combinations of heads/tails for 6 tosses!

We will place these probabilities into List 2.

5 Example: A coin is tossed 6 times. Construct the probability distribution for the number of tails in 6 tosses. By hand, this would be tedious since there are 64 combinations of heads/tails for 6 tosses! Let's construct it in the calculator! We need n, x, and p for this experiment. We will place these probabilities into List 2. X Values need to go into A P(X) values need to go into B Put x values 0 through 6 into A With dark gray box in B highlighted, press: menu, statistics, distributions, Binomial Pdf...(type in your n and p) The result is a probability distribution for the number of tails from 6 coin tosses.

EXAMPLE 2: The legendary Celtic small forward Larry Bird shot 88.6% from the free throw line over his entire NBA career. Suppose Bird attempts 8 free throws in one game. a. What proportion of FT attempts does Bird not make?

EXAMPLE 3: Ted Williams had 0.388 career batting average. Suppose he were to play in a game where he has 3 plate appearances. a. What is the probability he will go 2 for 3 in this game?

6 EXAMPLE 2: The legendary Celtic small forward Larry Bird shot 88.6% from the free throw line over his entire NBA career. Suppose Bird attempts 8 free throws in one game. a. What proportion of FT attempts does Bird not make? b. What is the probability he makes only 3 of his 8 attempts? c. How many do we expect Bird to make if he attempts 8? Remember, he makes 88.6% of his attempts. d. What is the chance Bird will make at most 3 of his 8 attempts? EXAMPLE 3: Ted Williams had career batting average. Suppose he were to play in a game where he has 3 plate appearances. a. What is the probability he will go 2 for 3 in this game? (He gets 2 hits in 3 attempts.) b. What is the expected number of hits out of 3 at bats? His batting average is c. Find the probability of Williams going 3 3 in this game. d. What is the probability of Williams getting at least 1 hit in 3 plate appearances?

EXAMPLE 4: You are a hospital manager and have been told that there is no need to worry that respirator monitoring equipment might fail because there is a 0.01 chance that any one monitor will fail.

7 EXAMPLE 4: You are a hospital manager and have been told that there is no need to worry that respirator monitoring equipment might fail because there is a 0.01 chance that any one monitor will fail. The hospital has 20 such monitors that work independently of each other. a. What is the probability that at least 1 of the 20 monitors fail? b. With which should you more concerned: one monitor failing or at least one monitor failing? Explain. EXAMPLE 5: Central Eye Clinic advertises that 90% of its patients approved for LASIK surgery to correct vision problems have successful surgeries. a. What is the expected number of successful surgeries out of 1000 LASIK procedures? b. What is the probability of having 10 or fewer unsuccessful surgeries for 100 procedures? Is this probability unusual? x, # unsuccessful Probability, P(x)

Statistics Chapter 8

Statistics Chapter 8 Binomial & Geometric Distributions Time: 1.5 + weeks Activity: A Gaggle of Girls The Ferrells have 3 children: Jennifer, Jessica, and Jaclyn. If we assume that a couple is equally