Section Conditional Probability Objectives: 1.Understand the meaning of conditional probability. 2.Learn the general Multiplication Rule:

Section Conditional Probability Example: The Sinking of the Titanic Do the data support the phrase “Women and children first?” 711/2201, or 32% of all passengers survived. 367/1731, or 21% of the males survived. 344/470, or 73% of the females survived. The chance of survival depends on the condition of whether the passenger was male or female. The notion that probability can change if you are given additional information is called conditional probability. Display 5.39 Gender MaleFemaleTotal Survived? Yes No Total

Section Conditional Probability Conditional Probability from the Sample Space Example: The Titanic and Conditional Probability Let S be the event that the passenger survived and let F be the event that the passenger was female. What is the probability of survival, given that the passenger was female? Display 5.39 Gender MaleFemaleTotal Survived? Yes No Total

Section Conditional Probability Conditional Probability from the Sample Space Example: Sampling Without Replacement When you sample without replacement from a small population, the probabilities for the second draw depends on the outcome of the first draw. Suppose you randomly choose two cards from a standard deck of 52 cards. Suppose the first card chosen is a heart. What is the probability that the second card chosen will also be a a heart?

Section Conditional Probability Conditional Probability from the Sample Space Example: Sampling Without Replacement When you sample without replacement from a small population, the probabilities for the second draw depends on the outcome of the first draw. Draw two cards from a standard deck of 52 cards. Suppose the first card chosen is a heart. What is the probability that the second card chosen will also be a heart? After the first heart is chosen, the sample space is changed. There are only 51 cards and 12 hearts remaining.

Section Conditional Probability Conditional Probability from the Sample Space D18. When you compare sampling with and without replacement, how does the size of the population affect the comparison? Conditional probability lets you answer the question quantitatively. Imagine two populations of students, one large (N = 100) and one small (N = 4), with half of each population male. Draw random samples of size n = 2 from each population. First, consider the small population. Find P(2nd is M | 1st is M), assuming you sample without replacement. Then calculate the probability again, this time with replacement.

Section Conditional Probability Conditional Probability from the Sample Space First, consider the small population (N = 4). Find P(2nd is M | 1st is M), assuming you sample without replacement. Then calculate the probability again, this time with replacement.

Section Conditional Probability Conditional Probability from the Sample Space Next, consider the large population (N = 100). Find P(2nd is M | 1st is M), assuming you sample without replacement. Then calculate the probability again, this time with replacement.

Section Conditional Probability Conditional Probability from the Sample Space Next, consider the large population (N = 100). Find P(2nd is M | 1st is M), assuming you sample without replacement. Then calculate the probability again, this time with replacement.

Section Conditional Probability Conditional Probability from the Sample Space How would you describe the effect of population size on the difference between the two sampling methods?

Section Conditional Probability Conditional Probability from the Sample Space How would you describe the effect of population size on the difference between the two sampling methods? If the population size is large relative to the sample size, sampling without replacement is about the same as sampling with replacement.

Section Conditional Probability The Multiplication Rule for P(A and B)

Section Conditional Probability The Multiplication Rule for P(A and B)

Section Conditional Probability The Multiplication Rule for P(A and B)

Section Conditional Probability The Multiplication Rule for P(A and B) Example:

Section Conditional Probability The Definition of Conditional Probability

Section Conditional Probability The Definition of Conditional Probability Example: Rolling Dice Find the probability that you get a sum of 8, given that you rolled doubles.

Section Conditional Probability Conditional Probability and Medical Tests Screening tests give an indication of whether a person is likely to have a particular disease or condition. A two-way table is often used to show the four possible outcomes of a screening test: Test Result PositiveNegativeTotal Disease Present aba+b Absent cdc+d Total a+cb+da+b+c+d

Section Conditional Probability Conditional Probability and Medical Tests The effectiveness of screening tests is judged using conditional probability.

Section Conditional Probability Conditional Probability and Medical Tests The effectiveness of screening tests is judged using conditional probability.

Section Conditional Probability Conditional Probability and Medical Tests A false positive is a positive test result when the patient does not have the disease or condition. A false negative is a negative test result when the patient has the disease or condition.

Section Conditional Probability E51. A screening test for the detection of a certain disease gives a positive result 6% of the time for people who do not have the disease. The test gives a negative result 0.5% of the time for people who do have the disease. Large scale studies have shown that the disease occurs in about 3% of the population. Fill in a two-way table showing the results expected for every 100,000 people. Test Result PositiveNegativeTotal Disease Yes No Total100,000

Section Conditional Probability E51. A screening test for the detection of a certain disease gives a positive result 6% of the time for people who do not have the disease. The test gives a negative result 0.5% of the time for people who do have the disease. Large scale studies have show that the disease occurs in about 3% of the population. Fill in a two-way table showing the results expected for every 100,000 people. Test Result PositiveNegativeTotal Disease Yes2, ,000 No6,00091,00097,000 Total8,50091,500100,000

Section Conditional Probability What is the probability that a person selected at random tests positive for this disease? What is the probability that a person selected at random who tests positive for the disease does not have the disease? Test Result PositiveNegativeTotal Disease Yes2, ,000 No6,00091,00097,000 Total8,50091,500100,000

Section Conditional Probability What is the probability that a person selected at random tests positive for this disease? What is the probability that a person selected at random who tests positive for the disease does not have the disease? Test Result PositiveNegativeTotal Disease Yes2, ,000 No6,00091,00097,000 Total8,50091,500100,000

Section Conditional Probability Conditional Probability and Statistical Inference To calculate a probability, you must work from a model. For example, what is the probability of observing an even number of dots on a roll of a die? The model is that the die is fair. Using this model, P(even|fair die) = 3/6 = 1/2 = Suppose you have a die that you suspect isn’t fair. How can you discredit the model that the die is fair? Suppose you roll the die 10 times and get an even number every time. P(10 evens) = (1/2) 10 = The outcome is so unlikely that the model of a fair die should be rejected.