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
Heads
0
1
2
Probability
0.25
0.50
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 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, non
Non, smoker
Smoker, non
Smoker, smoker
0.49
0.7 X 0.3 = 0.21
0.21
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.