Theoretical Probability Distributions

• Three major types:

• Binomial, Poisson, Normal

• Quick look at binomial and then concentrate on Normal Distribution

Binomial Distribution

• Based on events for which there are only 2 alternative possibilities:

• Heads or tails

• Girl or boy

• Pregnant or not

Multiple “Attempts”• The binomial distribution has the form:

• For one toss of the coin, what is the prob’y of a head

• For two tosses, what is the prob’y that both will be heads

• For three tosses, what is the prob’y that 3 will be heads

• For three tosses, what is the prob’y that 2 will be heads, etc.

What does “attempt” mean?

• Can mean individuals, patients, etc.

• Pregnant or not pregnant

• Out of the next 10 patients (10 attempts), what is the probability that 2 will be pregnant.

Exposed to a disease

• Get an infection or don’t get it

• If I have an accidental needle stab, what is the probability that I will contract Hepatitis B…….Let’s say it is 0.1

• What if I have 2 needle stabs, what is the probability that I will contract Hepatitis B.

Separate Distributions

• For one toss

• For two tosses

• For three tosses

• For four tosses

• (or one needle stab, or two patients, or rain on any of 3 days)

Ultra-Simple Distribution

For one toss

• P(0 heads) = 0.5

• P(1 head) = 0.5

Two Tosses

What’s the probability that one will be a head?

One out of two??? That would be 0.5

That’s correct in this case but have to be very careful. Really need some additional steps to show this.

CombinationsAll Possible Events for Two Tosses

• P(h, t)• P(h, h)• P(t, t)• P(t, h)

• Add up the P(1 head) = P(1 is a head and the other is a tail) = 2/4 = 0.5

Combinations cont’d

Look at another question. What’s the probability that neither will be a head?

That’s the same as saying:P(0) = P(both are tails) = 0.5 X 0.5 = 0.25

What’s the probability of at least one headP = 0.75 (Means the same as 1 – P(0).)

But…

• Before, we said that the probability of one head is 0.5

• What’s the difference?

• Ah…. Here we said, at least one head

• So, we mean probability of one head or two heads

Binomial Distribution

• Can be simple. You can put the data we just did into a table.

• Tail, tail = 0.25

• Head, tail = 0.25

• Tail, head = 0.25

• Tail, tail = 0.25

1.00

For two tosses

• P(0 Heads) = 0.25

• P(1, only 1) = 0.50

• P(2) = 0.25

Probability Distribution

Heads Probability0 0.251 0.502 0.25

What’s the probability of one or more heads:That is either one head or two so add the 2

possibilities = 0.75

More Complicated

• More than 2 attempts

• Or the probability of the single event might not be 0.5

Can always be done by the combinationsOr we use a formula

Another Factor in B.D., p

• In the first example, the probability that any one toss will be a head is taken to be 0.5

• Look at smoker or non- smoker. If we just consider undergraduate students at the current time, the probability of a smoker might be 0.3.

• What happens to the binomial distribution?

For two students

The same combinations

Non, non

Non, smoker

Smoker, non

Smoker, smoker

But what are the probabilities?

The same combinations

Non, non 0.49

Non, smoker 0.7 X 0.3 = 0.21

Smoker, non 0.21

Smoker, smoker 0.09

1.00

Factors in B.D.

• Number of “Attempts”

• Probability of any individual “success”

Binomial aka Bernoulli Distribution

• Just Make a Table of Combinations for the number of “attempts”

• And then pick out the probabilities

• Either add or multiply or subtract from 1, as needed.