CONDITIONAL PROBABILITY and INDEPENDENCE

1
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 odd1,
  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.

2
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.

3
Statistical independence, contd.

• Formally, events A and B are independent if and
  only if (iff )
• P(AB) P(A) or P(BA)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(AB)P(A and B)/P(B)P(A) and P(A and B)P(A) x
  P(B)

4
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.

5
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.