Basic Rules of Probability

1
Basic Rules of Probability

• Section 3.3

2
The Axioms

• If E is an event and is a subset of S, the sample
  space, then the following are true
• 0 P(E) 1 The probability of an event E is
  between 0 and 1 inclusive.
• P( ) 0 The probability of an empty set is
  zero. Consequence IF P(A n B) 0 then
• A n B which implies A and B are
  mutually exclusive.
• 3. P(S)1 The probability of the sample space
  is 1.

3
More Axioms

• 4. The rule for unions in general is
  P(A U B) P(A) P(B) P(A n B)
• 5. If A and B are mutually exclusive then
  P(A U B) P(A) P(B).
• 6.The rule for complimentary events is
  P(E) 1 P(E).

4
Examples

• Let E be the event of tossing two dice such that
  the sum of the face is even. Let F be the event
  that the sum of the faces is greater than 9.
• What is the probability of the sum of the faces
  being even and greater than 9.
• What we are looking for is P(E n F). The only
  rolls that correspond to that event are (6,4),
  (4,6), (5,5), (6,6).
• Thus P(E n F)4/361/9.
• What is the probability of the sum being even or
  greater than 9.
• Want to compute P(E U F).
• USE AN AXIOM P(E U F)P(E)P(F)-P(E n
  F)1/21/6-1/95/9

5
61 p.155

• Of all flashlights in a large shipment, 15 have
  a defective bulb, 10 have a defective battery,
  and 5 have both defects. If you purchase a
  flashlight from the shipment what are the
  probabilities of the following
• A defective light bulb or a defective battery,
• A good bulb or a good battery,
• A good bulb and a good battery.

6
ANSWERS

• L be the event of having a bad light bulb.
• B be the vent of having a bad battery.
• L is the event of having a good light bulb.
• B is the event of having a good battery.
• P(L)0.15, P(B)0.10, P(L n B)0.05
• The probability of having a bad bulb or bad
  battery is P(L U B)P(L)P(B)- P(L n B)
• 0.150.10-0.050.2020.
• The probability of having a good bulb or good
  battery is P(L U B).
• Using deMorgans law L U B(L n B)
• Thus P(L U B) P((L n B))1-P(L n B)1-0.05
  0.95 95
• The last part where we have a good bulb and good
  battery is P(LnB)P((L U B))1-P(L U
  B)1-.2.880