Download Conditional probability and independence Bernoulli trials and the

Conditional Probability and Independence
Example (You may view the table as giving numbers or percentages.)
men
Height ≤ x 24
Height > x 36
women
26
14
What is the probability that a randomly selected person is a man?
What is the probability that a randomly selected person is a woman?
What is the probability that a random person has height ≤ x?
What is the probability that a random man has height ≤ x?
What is the probability that a random person of height ≤ x is a woman?
Definition: If A and B are events, then the conditional probability of A given B is
P (A|B) =
P (A ∩ B)
P (B)
if P (B) is not zero, i.e., P (B) > 0.
Perspectives on conditional probability ...
Idea behind independence ...
Definition: Events A and B are independent if and only if P (A ∩ B) = P (A) · P (B).
Multiplication rules: Generally, P (A ∩ B) = P (A)P (B|A) = P (B)P (A|B).
If A and B are independent events, then P (A ∩ B) = P (A)P (B).
Law of Total Probability: If A1 , A2 , . . . are disjoint events and the event B ⊆ A1 ∪A2 ∪· · ·,
then
X
P (Aj )P (B|Aj ).
P (B) = P (A1 )P (B|A1) + P (A2 )P (B|A2 ) + · · · =
j
Bayes’s Formula: Under the hypotheses of the law of total probability, for each index k,
This follows immediately from
P (Ak )P (B|Ak )
P (Ak ) = P
.
j P (Aj )P (B|Aj )
P (Ak |B) =
and the law of total probability.
P (Ak ∩ B)
P (Ak )P (B|Ak )
=
P (B)
P (B)
Independence of more than two events: Surprisingly, there are situations in which
three or more events are independent of each other in pairs but are not independent of one
another more generally. See the bottom of page 24 for an example.
Events A1 , A2 , A3 , ... are independent if and only if the probability of the intersection of
every finite subcollection of these events is equal to the product of their probabilities.
Bernoulli Trials
The prototype of the Bernoulli trials process is repeated coin tossing. This is generalized in
the Bernoulli trials model.
1. In each trial there are two distinct possible outcomes: “success” or “failure.”
2. The probability of success p is the same in every trial.
(For convenience, let q = 1 − p, the probability of failure.)
3. The outcomes of the various trials are independent.
The Binomial Setting
If there is a fixed number n of Bernoulli trials, then we are in the “binomial setting.”
On page 17 our text shows that the probability of getting exactly k successes in n Bernoulli
trials is
!
n k
p (1 − p)n−k .
k
(In the text, θ = p.) Notice that, by the binomial theorem, these probabilities add up to 1:
n
X
k=0
!
n k
p (1 − p)n−k = (p + 1 − p)n = 1.
k