Let's take apart a typical problem about alarm clocks and see how the pieces fit together.

Puzzle Alarm Clock by evadedave

Working With Binomial Distributions

1. There are a set number of trials in this experiment. • In this case, there are 24 trials

A manufacturer produces 24 yard alarms per week. Six percent of all the alarms produced are defective. What is the probability of getting two defective alarms in one week?

The above problem is an example of a binomial probability experiment. Some properties of a binomial experiment are:

2. Each trial has exactly two possible outcomes: • alarm works • alarm fails

A manufacturer produces 24 yard alarms per week. Six percent of all the alarms produced are defective. What is the probability of getting two defective alarms in one week?

The above problem is an example of a binomial probability experiment. Some properties of a binomial experiment are:

3. events are independent. This means that the probability of success is the same for each trial. • i.e., each alarm system has a 6 percent probability of being defective.

A manufacturer produces 24 yard alarms per week. Six percent of all the alarms produced are defective. What is the probability of getting two defective alarms in one week?

The above problem is an example of a binomial probability experiment. Some properties of a binomial experiment are:

4. We will be looking for the probability of successes. • i.e. P(two defective alarms) = ?

A manufacturer produces 24 yard alarms per week. Six percent of all the alarms produced are defective. What is the probability of getting two defective alarms in one week?

The above problem is an example of a binomial probability experiment. Some properties of a binomial experiment are:

A manufacturer produces 24 yard alarms per week. Six percent of all the alarms produced are defective. What is the probability of getting two defective alarms in one week?

The above problem is an example of a binomial probability experiment. Some properties of a binomial experiment are:

5. The total of all probabilities is 1. This means that the alarm works or fails, and that no other possibility exists.

6. The data in a binomial problem are always discrete. • In this case, it means that the number of alarms is an integral value (1, 2, 3, 4, et cetera) which can be counted, and is not a continuous (i.e., measured) value, as is the case in a normal distribution

A manufacturer produces 24 yard alarms per week. Six percent of all the alarms produced are defective. What is the probability of getting two defective alarms in one week?

The above problem is an example of a binomial probability experiment. Some properties of a binomial experiment are:

A manufacturer produces 24 yard alarms per week. Six percent of all the alarms produced are defective. What is the probability of getting two defective alarms in one week?

'S' and 'F' (Success and Failure) are the possible outcomes of a trial in a binomial experiment, and 'p' and 'q' represent the probabilities for 'S' and 'F.'

• n = the number of trials• x = the number of successes in n trials• p = probability of success• q = probability of failure• P(x) = probability of getting exactly x successes in n trials

Note that 'Success' in this case, is the probability of selecting a defective alarm.

• P(S) = p • P(F) = q = 1 - p

A manufacturer produces 24 yard alarms per week. Six percent of all the alarms produced are defective. What is the probability of getting two defective alarms in one week?

'S' and 'F' (Success and Failure) are the possible outcomes of a trial in a binomial experiment, and 'p' and 'q' represent the probabilities for 'S' and 'F.'

• n = the number of trials• x = the number of successes in n trials, and 0 x n• p = probability of success• q = probability of failure• P(x) = probability of getting exactly x successes in n trials

Note that 'Success' in this case, is the probability of selecting a defective alarm.

• P(S) = p • P(F) = q = 1 - pSo how do we answer this question?

A manufacturer produces 24 yard alarms per week. Six percent of all the alarms produced are defective. What is the probability of getting two defective alarms in one week?

Now you try ...

Elaine is an insurance agent. The probability that she will sell a life insurance policy to a family she visits is 0.7 (she's a really GOOD sales lady).

(a) If she sees 8 families today, what is the probability that she will sell exactly 5 policies?

Now you try ...

Elaine is an insurance agent. The probability that she will sell a life insurance policy to a family she visits is 0.7 (she's a really GOOD sales lady).

(b) If she sees 8 families today, what is the probability that she will sell at most 5 policies?

http://www.math.uah.edu/stat/applets/BinomialCoinExperiment.xhtml

The Binomial Coin Experiment

binompdf(trials, p, x [this is optional]) trials = number of trials p = P(success) x = specific outcome

• When a manufacturer guarantees a toaster for one year, the toaster will work fine or fail in one year. The manufacturer knows that, on average, 5 percent of the toasters fail. What is the probability that the manufacturer will have to replace 120 or more toasters in a year when 2000 toasters are sold?

There are five children in a family. Assume that boys and girls are equally likely.

3. What is the probability that there are more than two girls?

2. What is the probability that there are at most three girls(i.e., there may be 0, 1, 2, or 3 girls)?

1. What is the probability that three are girls?

A shipment of 200 tires from a tire manufacturing company is known to include 40 defective tires. Five tires are selected at random, and each tire is replaced before the next tire is selected.

(b) What is the probability of getting at least 1 defective tire?

(a) What is the probability of getting at most 2 defective tires?

(c) What is the probability of getting 2 or 3 defective tires?

At a certain hospital, the probability that a newborn is a boy is 0.47. What is the probability that between 45 and 60 (inclusive) of the next 100 babies will be boys?

Normal Approximation to the Binomial Distribution

We have seen that binomial distributions and their histograms are similar to normal distributions. In certain cases, a binomial distribution is a reasonable approximation of a normal distribution. How can we tell when this is true?

Confused by flickr user slava

Recall:

In a normal distribution, we used values for μ and σ to solve problems, where:

• μ = the population mean, and • σ = the standard deviation

In a binomial distribution, we used values for 'n' and 'p' to solve problems, where:

• n = number of trials, and • p = probability of success

Normal Approximation to the Binomial Distribution

We now want to use the normal approximation of a binomial distribution. The distribution will be approximately normal if:

Once we know that a binomial distribution can be approximated by a normal curve we can calculate the values of μ and σ like this:

np ≥ 5 and nq≥ 5

Normal Approximation to the Binomial Distribution

this is the LINK

between these two

types of distributions

Link by flickr user jontintinjordan

An Example

Border patrol officers estimate that 10 percent of the vehicles crossing the US - Canada border carry undeclared goods. One day the officers searched 350 randomly selected vehicles. What is the probability that 40 or more vehicles carried undeclared goods?

Normal Approximation to the Binomial Distribution

What is n?

What is p?

What is q?

What is μ?

What is σ?

Is this binomial distribution approximately normal?

Is np ≥ 5?

Is nq ≥ 5?

Are the following distributions normal approximations of binomial distributions? How do you know?

(d) 80 trials where the probability of success on each trial is 0.99

(c) 600 trials where the probability of success on each trial is 0.05

(b) 60 trials where the probability of success on each trial is 0.20

(a) 60 trials where the probability of success on each trial is 0.05

Question #1HOMEWORK

Determine the mean and standard deviation for each binomial distribution. Assume that each distribution is a reasonable approximation to a normal distribution.

Question #2

(c) The probability of the Espro I engine failing in less than 50 000 km is 0.08. In 1998, 16 000 engines were produced. Find the mean and standard deviation for the engines that did not fail.

(b) 44 trials where the probability of failure for each trial is 0.28

(a) 50 trials where the probability of success for each trial is 0.35

A laboratory supply company breeds rats for lab testing. Assume that male and female rats are equally likely to be born.

(d) Is it correct to say that, in the above situation, P(x ≥ 120) = P(x > 119), or do we need to account for the values between 119 and 120?

(c) What is the probability that of 240 animals born, 120 or more will be female?

(b) What is the probability that of 240 animals born, 110 or more will be female?

(a) What is the probability that of 240 animals born, exactly 110 will be female?

Question #3Solve the following problem using a binomial solution