Probability Distributions

Discrete

Discrete data

• Discrete data can only take exact values

• Examples:• The number of cars passing a checkpoint in 30 minutes

• The show sizes of students in a class

• The number of tomatoes on each plant in a greenhouse

• Variables with many repeated values are treated as discrete

Continuous data

• Continuous data can be given values within a specified range or measured to a specified degree of accuracy.

• Examples:• The speed of a vehicle as it passes a checkpoint

• The mass of a cooking apple

• The time taken by a volunteer to perform a task

• Variables with few repeated values are treated as continuous

Binomial distribution

Attributes of a Binomial Experiment

• A binomial experiment is a statistical experiment that has the following properties:

Attributes of a Binomial Experiment

• The experiment consists of n repeated trials.

• Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.

• The probability of success, denoted by p, is the same on every trial.

• The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

When a coin is flipped, the outcome is either a head or a tail;

For convenience, one of the outcomes can be labeled "success" and the

other outcome "failure."

Two Outcomes

When a magician guesses the card selected from a deck, the magician can either be correct or incorrect;

Again for convenience, one of the outcomes can be labeled "success"

and the other outcome "failure."

Two Outcomes

When a baby is born, the baby is either born in the month of March or is not.

One of the outcomes can be labeled "success" and the other outcome

"failure."

Two Outcomes

In each of these examples, an event has two mutually exclusive possible outcomes.

One of the outcomes can be labeled "success" and the other outcome

"failure."

Two Outcomes

Consider the following statistical experiment.

• You flip a coin 2 times and count the number of times the coin lands on heads. This is a binomial experiment because

Consider the following statistical experiment.

• The experiment consists of fixed trials. We flip a coin 2 times.

• Each trial can result in just two possible outcomes - heads or tails.

• The probability of success is constant - 0.5 on every trial.

• The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials.

Experiment

• 3 dice- how many 4s?

• Does it meet the criteria of the Binomial?

Experiment

• The experiment consists of fixed trials. We rolled 3 dice

• Each trial can result in just two possible outcomes – ‘4’ or not a 4 on each dice

• The probability of success is constant – 1/6 on every trial (each dice).

• The trials are independent; that is, getting a ‘4’ on one trial does not affect whether we get a ‘4’ on other dice.

Let’s generalise it

Can you see the pattern?

Do these situations meet the conditions of the Binomial

distribution? • Experiment 1: A bag contains black, white and

red marbles that are selected one at a time, with replacement. The colour of each marble is noted.

• Experiment 2: This is a repeat of experiment 1 except that the bag contains black and white marbles only.

• Experiment 3: This is a repeat of experiment 1 except that the marbles are not replaced after each selection.

Do these situations meet the conditions of the Binomial

distribution? • At Mt Eden Foodtown, 60% of customers pay

by credit card. Find the probability that in a randomly selected sample of ten customers

• Exactly two pay by credit card • Fixed number of trials: 10 trials• Two outcomes: Pay by credit card or don’t• Probability remains constant: 60% (established

over a large number of transactions)• Independence: Randomly selected customers

Solution

• Number of ways of picking 2 out of 10 customers

Solution

• Number of ways of picking 2 out of 10 customers

Solution

• Number of ways of picking 2 out of 10 customers

Write out the answer in long form.

• At Mt Eden Foodtown, 60% of customers pay by credit card. Find the probability that in a randomly selected sample of ten customers

• More than seven pay by credit card

Write out the answer in long form.

• At Mt Eden Foodtown, 60% of customers pay by credit card. Find the probability that in a randomly selected sample of ten customers

• More than seven pay by credit card

Write out the answer in long form.

• At Mt Eden Foodtown, 60% of customers pay by credit card. Find the probability that in a randomly selected sample of ten customers

• More than seven pay by credit card

Graphics Calculator Binomial Dist

Stats Mode from Calc

F5 Distribution

Then F5 Binimial Dist

For point dist select Bpd

Select Data: Variable

For P(X=3), n = 12

p = 0.15For P(X=3) = 0.1720

For cumulative select Bcd

For P(X<3),

n = 10

p = 0.2

Since tables only go to 4 dp, round to 4dp

Graphics Calculator

• Notice that when using Bcd, you get the result that is less than or equal to the input number.

• If you needed >2, use 1 - (input 2)=

1 - 0.6778 = 0.3222

Using Tables

• 30% of pupils travel to school by bus.

• From a sample of ten pupils chosen at random, find the probability that

• Only three travel by bus.

Using Tables

• 30% of pupils travel to school by bus. From a sample of ten pupils chosen at random, find the probability that only three travel by bus.

• Fixed trials: 10 pupils• Two outcomes: travel by bus or don’t• Probability remains constant: 0.30• Independence: random selection of students

In general: The binomial probability for obtaining r successes in N trials is:

where P(r) is the probability of exactly r successes, N is the number of events, and π is the probability of

success on any one trial.

This formula assumes that the events:

• Number of trials is fixed

• fall into only two categories (2 outcomes which aremutually exclusive)

• Trials are independent (e.g. are randomly selected)

• Probability is the same for each trial

Consider this simple application of the binomial distribution: What is the probability of obtaining exactly 3 heads if a fair coin is

flipped 6 times? For this problem, N =6, r=3, and π =

0.5.Therefore,

Example 3

Two binomial distributions are shown below. Notice that for π = 0.5, the distribution is symmetric whereas for π = 0.3, the

distribution has a positive skew.

Example 4

• In a test there are ten multiple choice questions. For each question there is a choice of four answers, only one of which is correct. A student guesses the answers.

• Find the probability that he gets more than seven correct.

• He needs to obtain over half marks to pass and each question carries equal weight. Find the probability that he will pass.

Expectation and Variance

• Mean of the binomial

• Variance

The probability that it will be a fine day is 0.4. Find the expected number of fine days in a week and also

the standard deviation.

Answers are in days

The probability that a student is awarded a distinction in the mathematics examination is 0.05.

In a randomly selected group of 50 students, what is the most likely number of students awarded a

distinction?

• It is usually only necessary to consider the probabilities of values of X close to the mean.