Conditional Probability and Independence

1
Conditional Probability and Independence

• Section 3.6

2
Definition

• A conditional probability is a probability whose
  sample space has been limited to only those
  outcomes that fulfill a certain condition.
• The conditional probability of event A given that
  event B has happened is
• P(AB)P(A n B)/P(B).
• The order is very important do not think that
  P(AB)P(BA)! THEY ARE DIFFERENT.

3
Exercise 1

• Suppose that A and B are events with
  probabilities P(A)1/3, P(B)1/4,
• P(A n B)1/10
• Find each of the following
• 1. P(A B) P(A n B)/P(B)1/10/1/44/10
• 2. P(B A) P(A n B)/P(A)1/10/1/33/10
• 3. P(A B) P(A n B)/P(B)
• P((A U B))/(1-P(B))(1-P(A U B))/(1
  P(B))
• (1 (P(A)P(B)-P(A n B)))/(1-P(B))
• (1 (1/31/4-1/10))/(1-1/10)(1-29/60)/9/1
  0
• 31/60/9/1031/54.

4
Example, Using Table

• Let Ethe sum of the faces is even
• Let S2the second die is a 2
• Find 1. P(S2 E)
• P(S2 n E) /P(E)
• 3/181/6
• 2. P(E S2)
• 3/61/2

5
Example, Using Table

• One way of doing this is to construct a table of
  frequencies

Event E Event E TOTALS
Event S2 n(E n S2)3 n(E n S2)3 Total S2 18
Event S2 n(E n S2)15 n(E n S2)15 Total S2 18
Total E 18 Total E 18 Grand Total 36
6
Independence of events

• Two events E and F are said to be independent if
  and only if P(E n F)P(E)P(F).
• If the above condition is not satisfied, then we
  say the two events E and F are dependent.
• When we say two events are independent, we are
  saying that if event E has occurred, this will
  not effect the probability of event F.
• INDEPENDENT EVENTS The occurrence of one event
  has no effect on the probability of the other.

7
Independent Events

• Consider flipping a coin recording the outcome
  each time. Are these events independent????
• You throw 2 fair dice, one is green, one is red.
  Observe the outcomes.
• Let A be the event that the sum is 7
• Let B be the event that the red die shows an even
  number
• Are A and B Independent?
• Are A and B Mutually exclusive?