Download CONDITIONAL PROBABILITY and INDEPENDENCE

CONDITIONAL PROBABILITY and INDEPENDENCE
In many experiments we have partial information about the outcome,
when we use this info the sample space becomes smaller.
EXAMPLE. Roll a die. Events: A: score is odd={1, 3, 5}. B: score is 2.
C: score is 3
P(B)=1/6. Now, suppose we know A occurred. Then P(B given A)=0.
P(C)=1/6. Suppose A occurred, what is P(C)? The new sample space is
A ={1, 3, 5}. Then P(C given A)=1/3.
So, the conditional probability of C given A is the probability of C relative to
the probability of A. Formally:
P(C and A)
Probability of C given A = P(C |A)= ---------------- .
P(A)
Here, event A stands for the partial information, A is “the condition”.
Statistical Independence
Two events A and B are independent if the occurrence of one does not
affect the chances of the occurrence of the other.
EXAMPLES.
Toss 2 coins.
A: event that 1st comes up H;
independent events
B: event that 2nd comes up T.
Draw two cards from a deck without replacement.
A: event that 1st comes out red;
NOT independent events
B: event that second comes up red.
Statistical independence, contd.
Formally, events A and B are independent if and only if (iff )
P(A|B) = P(A) or P(B|A)=P(B).
Independence is a symmetric relation. If A is independent of B, then
B is independent of A.
Multiplication Rule. Events A and B are independent iff
P(A and B) = P(A) x P(B).
SUMMARY: For independent events A and B
P(A|B)=P(A and B)/P(B)=P(A) and P(A and B)=P(A) x P(B)
Statistical independence, contd.
Example. Toss two fair coins. Find probability that both coins come up
heads.
Solution. Since the results of the tosses are independent, by Multiplication
Rule:
P(H on 1st and H on 2nd)=P(H on 1st ) x P(H on 2nd)=(1/2) x (1/2)=1/4.
Example. Roll two fair dice. Find the probability that the score on the
first die will be odd and the score on the second die will be 5.
Solution. Since the dice are rolled independently:
P( odd 1st and 5 on 2nd)=P( odd on 1st ) x P(5 on 2nd)=(1/2) x (1/6)=1/12.
Notes on independence
NOTE 1: If A and B are independent, then A and (not B), (not A) and B,
and (not A) and (not B) are also independent. Thus:
P(A and not B) = P(A) x P(not B)=P(A) x (1-P(B));
P(not A and B)= P(not A) x P(B)= (1-P(A)) x P(B);
P(not A and not B)=P(not A) x P( not B) = (1 – P(A)) x ( 1- P(B)).
NOTE 2. If A and B and C are independent, then
P(A and B and C) =P(A) x P(B) x P(C),
that is Multiplication Rule extents to any number of events.