U NIT 5: P ROBABILITY —W HAT ARE THE C HANCES ? (P ULL OUT YOUR FORMULA SHEET )

What is the relationship between educational achievement and home ownership? A random sample of 500 people who participated in the 2000 census was chosen. Each member of the sample was identified as a high school graduate (or not) and as a home owner (or not). Overall, 340 were homeowners, 310 were high school graduates, and 221 were both homeowners and high school graduates.

A random sample of 500 people who participated in the 2010 census was chosen. Each member of the sample was identified as a high school graduate (or not) and as a home owner (or not). Overall, 340 were homeowners, 310 were high school graduates, and 221 were both homeowners and high school graduates.

(a) Create a two-way table that displays the data 500 people 340 homeowners 310 high school graduates 221 both homeowners and HS graduates.

Suppose we choose a member of the sample at random. Find the probability that the member (b) is a high school graduate. (c) is a high school graduate and owns a home. (d) is a high school graduate or owns a home.

5.3 C ONDITIONAL P ROBABILITY  Understand conditional probability and its relationship to independence

Conditional Probability The probability that an event will happen, given that we know that some other event has already happened

P(E F) Read as: The probability of E given that we know F has already happened Or just: The probability of E given F

Have a Tattoo? Totals YesNo Gender Male Female Totals

Type of Dwelling Single Family CondoMulti Family Loan Type Adjust Fixed

Rules of Probability: 1. 0 ⦤P(E)⦤ 1 2. P(S) = 1 3.If all events are equally likely, then P(E) = 4. P(E c ) = 1 − P(E) 5.P(E or F) = P(E) + P(F) − P(E and F)

6. If P(E ⧵ F) = P(E) then E and F are called independent Knowing the outcome of one event doesn’t provide additional info about the other event, just like in a scatterplot with no association.

Discussion of independence  Classical example using cards I draw one card at random from a standard deck. What is the probability that this card is a Jack? Oh, wait….I forgot to tell you….

Time to Defib Totals 2 min or less > 2 min Hospital size Small Med Large Totals

HardcoverPaperbackE-BookAudioTotals Fiction Non- Fiction

“Causes of Physician Delay in the Diagnosis of Breast Cancer” (Archives of Internal Medicine, 2002) Diagnosis Delayed Diagnosis Not Delayed Mamm Report Benign 3289 Mamm Report Suspicious 8304

5.3B C ONDITIONAL P ROBABILITY  Use the general multiplication rule

7.P(E and F) = P(E) ⋅ P(F ⧵ E) The General Multiplication Rule for the probability that two events both occur What would be true about this statement if E and F were independent? (The prob of E and F is the prob. of E times the prob. of F, given that E has occurred.)

Example: According to the last Census, 13.6% of the population is African-American. Only 4% of African-Americans are O-negative (the universal blood type). What percent of Americans are both African-American and have O-negative blood?

Organizers of a blood drive stress the importance of donations from members of the African American community. 10 African American students sign up for the THMS blood drive. What is the chance that at least one of these students is a universal donor?

A Formula for Conditional Probability 7. P(E and F) = P(E) ⋅ P(F ⧵ E) Let’s divide both sides by P(E)

The Scary Result of a Screening Test The probability for breast cancer for a woman of age forty who participates in routine screening is 1%. If the woman actually has breast cancer, the probability is 80% that she will get a positive result from the mammogram screening. If a woman does not have cancer, the probability is 9.6% that she will also get a positive result from the mammogram screening. A woman in this age group participates in a routine mammogram screening and gets a positive result. What is the probability that she actually has breast cancer?

Many employers require prospective employees to take a drug test. A positive result on this test indicates that the prospective employee uses illegal drugs. However, not all people who test positive actually use drugs. Suppose that 4% of prospective employees use drugs, the false positive rate is 5%, and the false negative rate is 10%. What percent of people who test positive actually use illegal drugs?

Additional Example: Lyme Disease is a tick-borne disease prevalent in the US and Europe. Approximately 0.207% of the population actually have Lyme Disease.

The false-positive rate of the blood test for Lyme disease is 3%. The false-negative rate is 6.3%

Imagine that you test negative for Lyme disease. What are the chances you actually are negative?

Looking for Oil When exploring for oil, surveyors look for leads (formations on the Earth’s surface that indicate the possibility of oil beneath the surface. Possibility does not mean oil is actually present. Suppose that only 6% of leads make good locations for oil wells. Oil company workers run tests on leads to determine the likelihood of the presence of oil. These tests are 95% accurate for both the presence and the absence of oil. If a lead does test positive, the company will drill an exploratory well to determine whether oil is present. What is the probability that a positive testing lead contains oil?

Additional Example: Approximately 1 in 3500 babies are born with a genetic disease called Cystic Fibrosis. “Woe to the child which when kissed on the head tastes salty.”

Before genetic testing, children were screened for CF using a sweat test. For simplicity’s sake, I estimate the false-positive rate of the sweat test to be approximately 2% and the false-negative rate to be approximately 1%

A child tests positive for CF. What is the approximate probability that this child has CF?