# Theoretical Probability Distributions Three major types: Binomial, Poisson, Normal Quick look at binomial and then concentrate on Normal Distribution.

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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.

Combinations All 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 head P = 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 HeadsProbability 00.25 10.50 20.25 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 combinations Or 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, non0.49 Non, smoker0.7 X 0.3 = 0.21 Smoker, non0.21 Smoker, smoker0.09 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.

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