### Baye`s Theorem

```Warm-up: A junk box in your room contains a dozen old
batteries, five of which are totally dead. You start picking
batteries one at a time and testing them. Find the following
probabilities.
• 1st two are both good
• At least one of the first 3 works
• The first four you pick all work
• You have to pick 5 batteries in order to find one that
works.
BAYE’S & AT LEAST
ONE
On January 28, 1986, Space Shuttle Challenger exploded on
takeoff. All seven crew members were killed. Following the
disaster, scientists and statisticians helped analyze what went
wrong. They determined that the failure of O-ring joints in the
shuttle’s booster rockets was to blame. Under the cold conditions
that day, experts estimated that the probability that an individual
O-ring joint would function properly was 0.977. But there were six
of these O-ring joints, and all six had to function properly for the
shuttle to launch safely. Find the probability that the shuttle would
launch safely under similar conditions.
What is the probability that in a family of 3 children there is
at least one girl?
Many people who come to clinics to be tested for HIV, the virus that
causes AIDS, don’t come back to learn the test results. Clinics
now use “rapid HIV tests” that give a result while the client waits.
In a clinic in Malawi, for example, use of rapid tests increased the
percent of clients who learned their test results from 69% to 99.7%.
The trade-off for fast results is that rapid tests are less accurate
than slower lab tests. Applied to people who have no HIV
antibodies, one rapid test has probability about 0.004 of producing
a false positive. What is the chance that at least one false positive
will occur?
Two mutually exclusive events can never be
independent!
• Let’s consider two mutually exclusive events.
• A – the person is male
• B- the person is pregnant
• Do you agree they are mutually exclusive?
• Independence:
Assumption of Independence
• If random sample of size n is taken from population of
size N, theoretical probability of successive selections
calculated with replacement and without replacement
differ by insignificant amounts when n is small compared
to N.
• Independence can be assumed if n is no larger than 5%
of N.
P(smoking) = .34
What’s the probability that 3 people smoke?
E1= new monitor works
E2=mouse works
E3=disk drive works
E4= processor works
P(E1)=0.98
P(E2) = 0.98
P(E3) = 0.94
P(E4) = 0.99
• P(Operates correctly) =
• P(all work except Monitor) =
A study found that 44% of college students engage in binge drinking, 37% drink
moderately, 19% abstain entirely. Another study found that among binge
drinkers 17% have been involved in an alcohol-related car accident, while
among non-bingers only 9% have been involved in such accidents. Find the
probability that a student who has an alcohol-related car accident is a binge
drinker.
Dan’s Diner employs three dishwashers. Al washes 40% of the dishes and
breaks only 1% of those he handles. Betty and Chuck each wash 30% of the
dishes, and Betty breaks only 1% of hers, but Chuck breaks 3% of the dishes he
washes. You go to Dan’s for supper one night and hear a dish break at the sink.
What’s the probability that Chuck is on the job?
Video-sharing sites, led by YouTube, are popular destinations on the Internet.
About 27% of adult Internet users are 18 to 29 years old, another 45% are 30
to 49 years old, and the remaining 28% are 50 and over. The Pew Internet and
American Life Project finds that 70% of Internet users aged 18 to 29 have
visited a video-sharing site, along with 51% of those aged 30 to 49 and 26% of
those 50 or older. What is the probability that an adult internet users visits
video-sharing sites? What is proportion of adult Internet users who visit
video-sharing sites are aged 18 to 29?
Box 1 has 2 red balls and 3 green. Box 2 has 4 red and 1 green. One ball
is randomly selected fromn Box 1 and put in box 2. What’s the probability
of a ball drawn from box 2 is green?
Suppose that a new Internet company Mumble .com requires all
employees to take a drug test. Mumble.com can afford only the
inexpensive drug test – the one with a 5% false-positive rate and a 10%
false-negative. Suppose that 10% of those who work for Mumble.com are
using the drugs for which Mumble is checking.
• Probability that the employee both uses drugs and tests positive.
• What is the probability that the employee does not use drugs but tests
positive anyway?
• What is the probability that the employee tests positive?
• If we know that a randomly chosen employee has tested positive, what
is the probability that he or she uses drugs?
Homework
```