|Conditional Probability Problems|
Algebra 2 pg. 744
Probability, Conditional Probability, Odds
RW: Finding a lost
On a recent Today show, two women were profiled who had known each other since childhood in England. After one of the women married, she moved away and the two women lost contact for 50 years. One day, while standing in line at a restaurant in California, the women struck up a conversation and each of them realized that she was speaking to her long-lost friend! After listening to their story, Katie Couric, the host of Today, pointed out that the chances of this happening must be ‘one in a million.’ One of the women, however, said that she believed the chances were probably closer to ‘one in a billion.’ Describe the steps you would take to compute the odds of this occurring. Be sure to list all of the factors that would have to be considered.
PT: Tree Diagrams, Conditional probability
19- year old Venus
Williams learned to play tennis at a public park in Compton,
California. She now competes
in professional tennis tournaments all around the world. On each point in tennis, a player is allowed two serves.
Suppose while playing tennis, Venus gets her first serve in, about
75% of the time. When she
gets her first serve in, she wins the point about 80% of the time.
If she misses her first serve, her second serve goes in about 90%
of the time. When this happens, she wins the point on her second serve
about 35% of the time.
a) Draw a tree diagram that represents all possible situations.
b) Find the probability that Venus wins a point when she is serving.
c) If you know she won a point while serving, what is the probability that she made her first serve?
Larson Algebra 1, p. 587
Finding missing money
You put $5.00 into the pocket of one of 6 pockets in your jeans, but you cannot remember which pocket. What is the probability that you find the money in the first pocket that you check? After checking two pockets without success, what is the probability that the $5.00 will be in the next pocket you check?
Glencoe Algebra 2, p. 204
PT: Conditional Probability
A weather station in a certain area gathers data about the chance of precipitation. It predicts that, if it rains on a given day, 50% of the time it will rain on the next day. If it is not raining it will rain on the next day only 30% of the time. The weather forecast for Monday predicts the chance of rain 80%. Find the chance of rain on Wednesday using this pattern.
Brown Pre-calculus, p.627
PT: Conditional Probability, Bayes’ Theorem
An auto insurance company charges younger drivers a higher premium than it does older drivers because younger drivers as a group tend to have more accidents. The company has 3 age groups: Group A includes those under 25 years old, 22% of all its policyholders. Group B includes those 25-39 years old, 43% of all its policyholders, Group C includes those 40 years old and older. Company records show that in any given one-year period, 11% of its Group A policyholders have an accident. The percentages for groups B and C are 3% and 2%, respectively.
a) What percent of the company’s policyholders are expected to have an accident during the next 12 months?
Suppose Mr. X has just had a car accident.
If he is one of the company’s policyholders, what is the
probability that he is under 25?
Say that this company not only classifies drivers by age, but in the case of drivers under 25 years old, it also notes whether they have had a driver’s education course. One quarter of its policyholders under 25 have had a drivers education course and 5% of these have an accident in a one-year period. Of those under 25 who have not had a driver’s education course, 13% have an accident within a one-year period. A 20-year-old woman takes out a policy with this company and within one year she as an accident. What is the probability that she did not have a driver’s education course?
Brown Pre-calculus, p. 628
PT: Conditional Probability, Bayes’ Theorem
A medical research lab proposes a screening test for a disease. To try out this test, it is given to 100 people, 60 of whom are known to have the disease and 40 of whom are known not to have the disease. A positive test indicates the disease and a negative test indicates no disease. Unfortunately, such medical tests can produce two kinds of errors:
1) A false negative test: For the 60 people who do have the disease, this screening indicates that 2 do not have it.
2) A false positive test: For the 40 people who do not have the disease, this screening test indicates that 10 do have it.
a) Which of the false tests do you think is more serious and why?
b) Incorporate the facts given above into a tree diagram. (Be sure to convert the given integers into probabilities.)
Suppose the test is given to a person whose disease status is
unknown. If the test result
is negative, what is the probability that the person does not have the
a) What is the probability that the diagnostic test gives:
1) A false negative result?
2) A false positive result?
b) What is the probability that:
1) The diagnostic test gives the correct result?
2) A person with a positive diagnostic test has the disease?
A person with a negative diagnostic test does not have the disease?
Key to Problem Bank: