Transcript Section 5.3 Simulations

Warm-up – for my history
buffs…
 A general can plan a campaign to fight
one major battle or three small battles.
He believes that he has probability 0.6 of
winning the large battle and probability of
0.8 winning each of the small battles.
Victories or defeats in the small battles
are independent. The general must win
either the large battle or all three small
battles to win the campaign. Which
strategy should he choose?
Wrapping Up
Chapter 5
Tree Diagrams
Tree Diagrams
 Suppose that 2% of a clinic’s patients are known to be
HIV+. A blood test is developed that is positive in 98%
of patients with HIV, but is also positive in 3% of
patients without HIV.
 Find P(positive test).
 Find P(positive test ∩ HIV+)
 If a person who is chosen at random from the clinic’s
patients is given the test and it comes out positive,
what is the probability that the person actually has
HIV?
Decisions, decisions!
 Many probability problems involve
making a decision. A tree diagram can
help us organize the information.
Dialysis or Transplant?
 Lynn has to decide between dialysis or a kidney
transplant. Here are the facts:
 52% of dialysis patients survive for 3 years.
 After 1 month, 96% of kidney transplants succeed. 3% fail to
function, and 1% die. Patients who return to dialysis still have
a 52% chance of surviving 3 years.
 Of the successful transplants, 82% continue to function for 3
years. 8% must return to dialysis, of whom 70% survive to the
3 year mark. 10% of the successful transplants die without
returning to dialysis.
Example
 The probability of rain today is .3. Also, 40% of
all rainy days are followed by rainy days and
20% of all days without rain are followed by rainy
days. The following tree diagram represents the
weather for today and tomorrow.
 Complete a tree diagram to represent this situation.
 What is the probability that it rains on both days?
 What is the probability that it rains on one of the two
days?
 What is the probability that it does not rain on either
day?
Testing for Independence
 Remember the general rule for
multiplication:
 P(A∩B) = P(A)*P(B|A)
 Also remember the multiplication rule for
independent events:
 P(A∩B) = P(A)*P(B) if A and B are
independent.
There are two ways to
test for independence:
 P(A∩B) = P(A)*P(B|A)
 If A and B are independent, then P(A∩B)
= P(A)*P(B)
 Therefore, by substitution, if A and B are
independent, then P(B|A) = P(B)
Here is the table of Central
High’s student population.
Seniors
Juniors
Sophomores
Total
Male
156
168
196
520
Female
144
172
164
480
Total
300
340
360
 a. What is the probability of selecting a male?
 b. What is the probability of selecting a male
from the sophomore class?
 c. Use your answers from parts a and b to
determine whether the events “selecting a
male” and “selecting a sophomore” are
independent.
Homework
Chapter 5 # 63, 71, 73, 83, 85, 88
HW answers

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


63. a) .2428
b) .5983
71. a) .41
b).6341
73. b < b/t < t < t/b
85. .1423
88. a) diagram
b) .6387