Binomial Distributions



AP Stats / CH 8: Binomial vs Geometric Distributions

Distribution

• Every variable has a distribution, whether the variable is categorical or numerical, continuous or discreet.

• A variable’s distribution consists of the values it takes on and how often it takes each of these values.

• Distributions are often shown as graphs so you can see the relative frequency with which values occur. Tables are used, too.

Properties of a binomial distribution: #1-4 are the 4 conditions that indicate a binomial distribution (a class of discrete probability distributions)

• 1) There are n repeated trials – a fixed number of observations

• 2) All trials are identical and independent

• 3) The probability of success is the same for each observation

• 4) Each trial has exactly two possible outcomes, “success” and “failure”

•

• P(success in a given trial) = p

• P(failure in a given trials) = 1 – p

• If X = the number of successes in n trials = 0, 1, 2, …, n

• X is B(n, p) ----( n observations, probability p, k successes

• P(X=k) = [pic] …..

• Mean of the binomial random variable: [pic]

• Variance of the binomial random variable: [pic]

• TI83: binompdf(n, p, X) -- under Distr.

• (pdf = probability distribution function)

• TI89: tistat.binomPdf(n, p, X) ---in CATALOG under Flash Apps

• P(X [pic]n) = sum of each probability from n and below

• P(X>n) = 1 – P(X [pic]n)

almost binomial (a.k.a. binomial setting)

As a rule of thumb, we consider a situation to be almost binomial when the population is at least 20 times the sample size. (Some textbooks say 10 times the sample size.) In terms of statistical analysis, almost binomial situations can be treated as though they are binomial.

Binomial --(Normal

“As the number of trials n gets larger, the binomial distribution gets closer to the normal distribution.” When n is large and the binomial calculations are cumbersome, we can use normal probability calculations to approximate the binomial probabilities.

Rule of thumb: You can use normal approx. if :

[pic]

Properties of a geometric distribution:

• the trials are repeated until the first success is observed

• All trials are identical and independent

• Each trial has two possible outcomes, “success” and “failure”

• P(success in a given trial) = p

• P(failure in a given trials) = 1 – p

• [pic]

• [pic]

• P(X>n) = [pic] (prob. that it takes more than n trials for a success)

Binomial or geometric or neither?

1. A worker opening oysters to look for pearls counts the number of oysters he has to open until he finds the first pearl.

2. A supervisor at the end of an assembly line counts the number if non-defective items produced until he finds a defective one.

3. Deal 10 cards. Count the number X of red cards.

4. A quality control inspector takes a random sample of 20 items from a large lot, inspects each item, classifies each as defective or non-defective, and counts the number of defective items in the sample.

5. An engineer chooses and SRS of 10 switches from a shipment of 10,000 switches. Suppose 10% of the switches in the shipment are bad. The engineer counts the number X of bad switches in the sample.

6. A homeowner buys 30 azalea plants from a nursery. The number if plants that survive at the end of the year are counted.

7. An electrician inspecting cable one yard at a time for defects counts the number if yards he inspects before he finds a defect.

8. A game requires you to roll a pair of dice until you get doubles.

9. A teacher asks you to roll a pair of dice 10 times and record the number of times you got doubles.

BINOMIAL

Ex. 1 Corinne is a basketball player who makes 75% of her free throws over the course of a season. In a key game, Corinne shoots 12 free throws and makes only 7 of them. Is this unusual for Corinne?

The # of X baskets in 12 attempts has the B(12, 0.75) distribution.

Find P(X=7)

Find P(X [pic]7)

EX. 2 Each child born to a particular set of parents has probability 0.25 of having Blood type O. Suppose these parents have 5 children and we are interested in the number of children who have type O blood. Thus, X is B(5, 0.25).

a) What is the probability that exactly 2 of the children have type O blood?

EX. 3 An SRS of 10 switches from a shipment of 10,000 switches as probability .10 of being bad.

a) Find the probability that no more than 1 switch is bad.

b) Find the mean and standard deviation.

Ex. 4. In a recent survey 2500 people were asked if shopping has become frustrated. Suppose that in actuality 60% of US adults “Agree”.

a) What is the probability that 1520 or more of the sample agree?

EX. 5 Suppose a family eats at a fast food restaurant. There are three possible toys (a car, a top, or a yo-yo) given with the kids’ meals. The toys are placed in the meal bags at random. Suppose this family buys a kid’s meal at this restaurant on four different days.

a. What is the probability that 3 out of 4 meals will come with a yo-yo?

b. What is the probability that at most 2 meals will come with a yo-yo?

c. What is the probability of getting at least 3 yo-yos with 4 meals purchased?

d. What is the expected number of yo-yos when 4 meals are purchased?

e. How can we simulate this process?

f. Let’s repeat it 100 times and record the results.

Mean of the geometric random variable: [pic]

Variance of the geometric random variable: [pic]

g. How can we turn this kids meal problem into a geometric distribution?

h. Find the probability that the family will have to buy 3 meals to get a yo-yo.

i. Find the probability that the family will have to buy 5 meals to get a yo-yo.

j. What is the expected number of meals needed to get a yo-yo?

k. Let’s simulate this problem using a six sided die.

l. Repeat the process 100 times. Record the results.

A probability experiment is an experiment whose possible outcomes may be known but whose exact outcome is a random event and cannot be predicted with certainty.

Binomial Distributions

In order to use a sample value to make an inference about a population value, you use a probability distribution. A probability distribution tells you whether your sample value is common, or whether it's a value you'd almost never expect to get. If you find that your sample value is extremely rare according to your probability distribution, you may have reason to doubt some assumptions you made about the population.

This lesson looks at the binomial distribution, a probability distribution useful for making many types of inferences. This distribution is useful when modeling the likelihood of events either succeeding of failing independently, such as voters voting for or against a ballot measure. Under certain conditions, you can use the normal distribution as a model for the binomial distribution. This comes in handy, since you can use z-scores to find probabilities like you'd do with a normal distribution.

Objectives

By the end of this lesson you should be able to:

• List the four conditions that define a binomial setting, and use those conditions to determine whether a situation is binomial.

• Solve binomial problems involving at least, at most, and between.

• Use the normal approximation in binomial setting problems.

APEX Notes

binomcdf / binomial probabilities for a range of outcomes

Binomial cumulative density function. The function on the TI-83 calculator that finds the probability for a range of successes within n trials.

The syntax is binomcdf(n,p,x), where n is the number of trials, p is the probability of success for each trial, and x is the upper bound for the range of successes.

For example, binomcdf(20,.75,8) will find the probability of getting 8 or fewer successes in a situation with 20 trials and a probability of success for each trial of .75. Access this function by pressing 2nd [DISTR], and scrolling down to binomcdf(.

Note that since the binomial distribution is discrete, P(x < 4) would have an upper bound of 3 (you are not including the 4), but P(x[pic] 4) would have an upper bound of 4 (since you would include the 4). Therefore, if you want to calculate P(X < 4), the syntax is binomcdf(20,.75,3).

binomial experiment

When we count the number of successful outcomes in a binomial event, we refer to this as a binomial experiment.

binomial probabilities for a range of outcomes

The probability of having a range of outcomes rather than a single value. For example, the probability of having less than or equal to x successes out of n trials from a binomial distribution B(n, p). The formula for calculating such a probability is P(X x) = P(X=1) + P(X=2) + ... + P(X=x). In order to do these computations, we most often use the binomcdf(n,p,x) function found in the DISTR menu of the TI-83.

binompdf / binomial probabilities for an exact number of outcomes

Probability of having exactly x successes out of n trials from a binomial distribution B(n, p) The formula for calculating such probability is P(X=x)=[pic] p x (1-p) n-x, X=0,1, ... , n,where [pic]gives the number of ways to select x things from n things. The TI-83 computes this as binompdf(n,p,x), which can be found in the DISTR menu of your calculator.

Binomial probability density function.

The function on the TI-83 calculator that finds the probability for an exact number of successes within n trials. The syntax is binompdf (n,p,x), where n is the number of trials, p is the probability of success for each trial, and x is the number of successes. Access this function by pressing 2nd [DISTR], and scrolling down to binompdf(.

Continuity correction

When using the normal approximation to the binomial, you can increase accuracy by treating values as though they include the .5 below and above them. In other words, include .5 below the lower bound, and .5 above the upper bound. Be sure to use the true upper or lower bound. For instance, if you are looking for P(x < 4), your upper bound is 3 because your interval does not include 4. So to use the continuity correction you would find P(x 2.5) If you are looking for P(x 4), then your upper bound does include 4, and to use the continuity correction you would find P(x 3.5),

Note that if your upper or lower bound is infinity, you only need to add .5 to the bound that is not infinity.

binomial setting

A binomial setting has the following properties:

1. There is a set number, n, of identical trials

2. The outcome of each trial is either a success or a failure.

3. The probability of success on a single trial is p.

4. The trials are independent. In a binomial experiment, you are interested in the probability of getting x successes in n trials.

geometric setting

A geometric setting has the following properties: The trials are independent. The outcome of each trial is either success or failure. Probability of success on a single trial is p.

normal approximation to the binomial

If X follows the B(n, p) distribution, and if np 10 and n(1 - p) 10 many textbooks use 5 instead of 10), then X follows an approximately normal distribution, with mean and standard deviation: [pic]

Binomial Situations (Events)

By now in your study of statistics, you should be able to answer such questions as, What's the probability of getting exactly three heads on five flips of a coin? or What's the probability of getting a value between 8 and 10 from a normal distribution with a mean of 11 and a standard deviation of 2?

average waiting time Average number of trials until you get the first success. For a geometric setting, average waiting time is always 1/p.

geometcdf / Geometric cumulative density function.

The function on the TI-83 calculator that finds the probability in a geometric setting of getting the first success in x trials or less. The syntax is geometcdf(p,x), where p is the probability of success for each trial, and x is the upper bound for the range of the number of trials. For example, geometcdf(.75,8) will find the probability of getting your first success in 8 trials or less, if the probability of success on each trial is .75.

Access this function by pressing 2nd [DISTR], and scrolling down to geometcdf(.

Note that since the geometric distribution is discrete, P(x < 4) has an upper bound of 3 (you do not include the 4), but P(x [pic]4) has an upper bound of 4 (since you include the 4). Thus if you want to calculate P(X, 4), the syntax is geometcdf(.75,3).

geometric distribution / geometric setting

A distribution of the number of trials required to obtain the first success under a geometric setting. Suppose X is a random variable with a geometric distribution and has probability of success p on each trial. Then X takes the values 1, 2, 3, ..., and P(X = n) = (1 - p)n-pp.

When Are You Most Likely to Get Your First Red Candy?

Discussion Topic

Do you recall the activity in class in trying to get out of jail when playing Monopoly? To get out, you need to roll doubles in three tries or fewer or you have to pay. On average, how many times would people have to roll before getting doubles, and is that number larger than three? This is an average waiting-time question, where you're interested in the number of tries you need on average to get the outcome you want.

You'll learn more about situations like this and the probability distributions they produce in the next Tutorial. Meanwhile, this Discussion will help you think about waiting-time situations.

Offer a response to any one (or more) of the following, or respond to another student. As you explain your reasoning, be sure to use what you know about the laws of probability.

1. If you buy a very large bag of candies colored brown, yellow, green, blue, orange, and red, and you start eating them, how many candies would you expect to pick until you got a blue one? Explain your reasoning using what you know about probability.

2. What's the probability you'll get out of Jail in Monopoly without having to pay? In other words, what are the chances you'll roll doubles on two six-sided dice within three tries?

3. You may be familiar with promotional campaigns where a company's products are marked with a letter, under the bottle cap of a soft drink, for example, and you're supposed to spell something to win a prize. In these cases, do you think some game pieces are more common than others? Describe an experiment you could conduct to see if some game pieces are easier to get than others.

4. Say you're playing a game like the one described in topic 3. A soft drink company has a letter printed on each bottle cap and the object is to spell the words I bought a lot of bottles to spell this. You have all the letters you need except p. The company's disclaimer statement says for each bottle you buy there's a 1/200 chance of getting a p. On average, how many bottles would you expect most people to buy in order to get this letter?

5. Create your own question about an average = waiting-time situation, or describe a real one you've seen or participated in. Explain why it's an average = waiting-time situation, and invite other students to answer it.

6. Have you ever won anything in a game like the ones described in topics 3 and 4? Describe the game, and calculate the probability of winning.

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