Chapter 4 Probability: Probabilities of Compound Events

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Chapter 4 Probability Probabilities of Compound
Events

• 4.1THE ADDITION RULE
• 4.1.1 The General Addition Rule
• 4.1.2The Special Addition Rule for Mutually
  Exclusive Events
• 4.2 Conditional Probabilities
• 4.3 The Multiplication Rule
• 4.4 Independent Events and the Special
  Multiplication Rule
• 4.4.1Independence of Two Events
• 4.4.2Independence of More Than Two Events and the
  Special Multiplication Rule
• 4.5 Bayes Theorem
• 4.5.1 The Total Probability
• 4.5.2 Bayes Theorem

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4.1THE ADDITION RULE

• 4.1.1 The General Addition Rule
• Example 1
• Events A and B are such that P(A) 19/30 , P(B)
  2/5 and P(A?B)4/5 . Find P(A?B).
• (Ans 9/30)

The general addition rule for two events, A and
B, in the sample space S P(A?B) P(A) P(B)
P(A?B)
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• Example 2
• In a group of 20 adults, 4 out of the 7 women and
  2 out of the 13 men wear glasses. What is the
  probability that a person chosen at random from
  the group is a woman or someone who wears
  glasses? (Ans 1/5)
• Example 3
• A class contains 10men and 20 women of which half
  the men and half the women have brown eyes. Find
  the probability p that a person chosen at random
  is a man or has brown eyes. (Ans 2/3)

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• The General Addition Rule for Three Events

P(A?B?C) P(A) P(B) P(C) P(A?B) P(A?C)
P(B?C) P(A?B?C)
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• 4.1.2 The Special Addition Rule for Mutually
  Exclusive Events
• Example 1
• Records in a music shop are classed in the
  following sections
• classical, popular, rock, folk and jazz. The
  respective probabilities that a customer buying a
  record will choose from each section are 0.3,
  0.4, 0.2, 0.05 and 0.05. Find the probability
  that a person (a) will choose a record from the
  classical or the folk or the jazz sections, (b)
  will not choose a record from the rock or folk or
  classical sections.

If A1, A2, , Ak are mutually exclusive,
then P(A1?A2??Ak) P(A1) P(A2) P(Ak).
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4.2 Conditional Probabilities
If A and B are two events and P(A) ? 0 and P(B) ?
0, then the probability of A, given that B has
already occurred is written P(AB) and P(AB)

• Example 1
• Given that a heart is picked at random from a
  pack of 52 playing cards, find the probability
  that it is a picture card.

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• Example
• When a die is thrown, an odd number occurs. What
  is the probability that the number is prime?
• Example
• Two tetrahedral, with faces labelled 1,2,3 and 4,
  are thrown and the number on which each lands is
  noted. The score is the sum of these two
  numbers. Find the probability that the score is
  even, given that at least one die lands on a 3.

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4.3 The Multiplication Rule
The general multiplication rule for events A and
B in the sample space S P(A?B) P(A) P(BA)
P(A?B) P(B) P(AB)
P(A?B?C) P(A) P(BA) P(CA?B)
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4.4 Independent Events and the Special
Multiplication Rule

• 4.4.1 Independence of Two Events
• Note If two evens are mutually exclusive, then
  P(A?B) _______. So for two events to be both
  independent and mutually exclusive we must have
  P(A) P(B) P(A?B) ________. This is possible
  only if either P(A) _________ or P(B)
  __________.

If the occurrence or non-occurrence of an event A
does not influence in any way the probability of
an event B, then event B is independent of event
A and P(BA) P(B).
Two events A and B are independent iff P(A?B)
P(A)P(B)
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• Example 1
• A die is thrown twice. Find the probability of
  obtaining a 4 on the first throw and an odd
  number on the second throw.
• Example 2
• A bag contains 5 red counters and 7 black
  counters. A counter is drawn from the bag, the
  colour is noted and the counter is replaced. A
  second counter is then drawn. Find the
  probability that the first counter is red and the
  second counter is black.
• Example 3
• A fair die is thrown twice. Find the probability
  that (a) neither throw results in a 4, (b) at
  least one throw results in a 4.
• Example 4
• Two events A and B are such that P(A) , P(AB)
  and P(BA) .
• Are A and B independent events? (b) Are A and B
  mutually exclusive events?
• (c) Find P(A?B). (d) Find P(B).

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• 4.4.2 Independence of More Than Two Events and
  the Special Multiplication Rule

If k events A1, A2,., Ak are independent,
then P(A1 ? A2 ?.? Ak) P(A1)P(A2)P(Ak)
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• Example 1
• A die is thrown four times. Find the probability
  that a 5 is obtained each time.
• Example 14
• Three men in an office decide to enter a marathon
  race. The respective probabilities that they will
  complete the marathon are 0.9, 0.7 and 0.6. Find
  the probability that at least two will complete
  the marathon. Assume that the performance of each
  is independent of the performances of the others.

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• C.W
• Conditional Probability
• 1)
• In a family of two children with at least one
  girl. What is the probability that the other one
  is a boy?
• 2)
• Suppose a box contains 3 white balls and 5 red
  balls.
• Balls are drawn randomly one by one without
  replacement from it. What is the probability that
  the second ball drawn will be red, given that the
  first ball drawn is white?
• Balls are drawn randomly one by one with
  replacement from it. What is the probability that
  the third ball drawn will be white, given that
  the first two balls drawn are white.

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• 3)
• A credit card company has surveyed new accounts
  from university students. Suppose a samples of
  160 students indicated the following information
  in terms of whether the student possessed a
  credit card X and/or a credit card Y.

credit card X credit card X
credit card Y Yes No
Yes 50 20
No 30 60
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• 4.) Let event A students possessed two
  credit cards.
• event B students possessed at least one credit
  card.
• event C students did not possess any card.
• event D students possessed a credit card X.
• event E students possessed a credit card Y.
• Find the probabilities of each of these events
  A,B,C,D,E,
• Find also and .

. Find also
,
.
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• 5) A fair coin is tossed three times
• Let event A Head appears on first toss.
• event B Head appears on second toss.
• event C Head appears on all three tosses.
• To find whether A and B, B and C, C and A are
  independent.

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4.5 Bayes Theorem

• 4.5.1 The Total Probability

Suppose a sample space S is partitioned into k
mutually exclusive events Ej (j 1,2,,k), i.e.
S E1?E2?.?Ek with Ei?Ej ? for i?j,
then P(A) P(E1)P(AE1) P(E2)P(AE2)
P(Ek)P(AEk)
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• 4.5.2 Bayes Theorem

Let the sample space S be partitioned into
mutually exclusive events Ejs (j 1,2,,k) and
let A be an event in S. Then the probability of
Er conditional on A is P(Er A)
for r 1,2,,k
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• Suppose there are three identical boxes which
  contain different number of white and black
  balls.
• A box is selected at random and a ball is drawn
  from it randomly .
• (I) What is the probability that a white ball
  is chosen?
• (ii) Suppose a white ball is chosen, find the
  probability that this white ball comes from the
  1st box.

Number of white balls Number of black balls
1 st box 8 3
2 nd box 6 5
3 rd box 4 7
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• 2) The marketing manager of a soft drink
  manufacturing firm is planning to introduce a new
  rand of Coke into the market. In the past, 30
  of the Coke introduced by the company have been
  successful, and 70 have not been successful.
  Before the Coke is actually marketed, market
  research is conducted and a report, either
  favorable or unfavourable, is compiled. In the
  past, 80 of the successful Coke received
  favourable reports and 40 of the unsuccessful
  Coke also received favourable reports. The
  marketing manager would like to know the
  probability that the new brand of Coke will be
  successful if it receives a favourable report.

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• 3)
• A man decided to visit his friend at North Point.
  He can reach there by MTR, Bus or Tram
  respectively. The following information is given
• (i) He was late for his visit. Find the
  probability that he had travelled by MTR.
• (ii) He was not late for his visit. Find the
  probability that he had travelled by Bus.

Probability of being taken Probability of being late
MTR 5/8 1/4
Bus 2/8 5/9
Tram 1/8 7/8