Probability. Esra Akdeniz. February 26, 2016

Transcription

1 Probability Esra Akdeniz February 26, 2016

2 Terminology An experiment is any action or process whose outcome is subject to uncertainty. Example: Toss a coin, roll a die. The sample space of an experiment is the set of all possible outcomes of that experiment. Example: Roll a die. An event is any collection of outcomes contained in the sample space S. The union of two events A and B All outcomes in at least one of the events, A OR B. The intersection of two events A AND B. The complement of an event A, A or A c.

3 Probability The probability of an event A is its relative frequency of occurrence or the proportion of times the event occurs in a large number of trials repeated under identical conditions. The probability of the event A is denoted by P(A), which will give a precise measure of the chance that A will occur.

4 Axioms of Probability Axiom 1: For any event A, P(A) 0. Axiom 2: P(S) = 1. Axiom 3: If A 1, A 2, A 3,... is an infinite collection of disjoint events, then P(A 1 A 2 A 3...) = P(A i ). i=1

5 Probability Properties P( ) = 0, where is the null event. For any event A, P(A) = 1 P(A ). For any events A and B, P(A B) = P(A) + P(B) P(A B). Extension for three events A, B and C: P(A B C) = P(A)+P(B)+P(C) P(A B) P(A C) P(B C)+P(A B C).

6 Example Employment Status Population Hearing Impairments due to injury Currently employed 98, Currently unemployed Not in the labor force 56, Total 163, Table: Data collected in the National Health Interview in the U.S. between years E 1: currently employed. P(E 1) = E 2: currently unemployed. P(E 2) = E 3: is not in the labor force. P(E 3) = H: hearing impairment due to injury. P(H)= Probability of an individual having a hearing impairment and being unemployed. Probability of an individual having a hearing impairment and being employed.

7 Conditional Probability Conditional Probability is the probability that an event will occur when we know some facts that have already happened. Notation: Conditional Probability of event A given that the event B has occured: P(A B)

8 The Multiplicative Rule of Probability P(A B) = P(A B) P(B)

9 Example Employment Status Population Hearing Impairments due to injury Currently employed 98, Currently unemployed Not in the labor force 56, Total 163, E 1: currently employed, E 2: currently unemployed, E 3: is not in the labor force, H: hearing impairment due to injury. Find the probability of an individual having a hearing impairment given that s/he is unemployed. employed. not in the labor force. Probability of an individual being unemployed given that s/he has hearing impairment.

10 The Law of Total Probability Let A 1,..., A k be mutually exclusive and exhaustive events. Then for any event B, P(B) = k P(B A i )P(A i ) i=1 The events A 1,..., A k are said to be exhaustive if one of them must occur, that is A 1... A k = S.

11 Bayes Theorem Let A 1,..., A k be mutually exclusive and exhaustive events with P(A i ) > 0 for i = 1, 2,..., k. Then for any event B for which P(B) > 0, j = 1, 2,..., k. P(A j B) = P(A j B) P(B) = P(B A j )P(A j ) k i=1 P(B A i)p(a i )

12 Microchips from a factory are sorted into three separate boxes. Box 1 contains 25 microchips from shift 1, box 2 contains 35 microchips from shift 2, and box 3 contains 40 microchips from shift 3. There are 5, 10 and 5 defective microchips in the first, second and third boxes, respectively. Let A denote the event that a defective microchip is obtained and B 1, B 2 and B 3 be the events of choosing box 1, box 2 and box 3, respectively. What is the probability of obtaining a defective microchip? If we picked a defective microchip, what is the probability that is from box 1?

13 Example In 1992, there were 4,065,014 registered births in the United States. Of these infants, 2,081,287 were boys and 1,983,727 were girls. B: a child will be a boy, P(B). G: a child will be a girl, P(G). P(B G) and P(B G).

14 Example (Cont d) In 1992, there were 4,065,014 registered births in the United States. Of these infants, 2,081,287 were boys and 1,983,727 were girls. We select two women. B 1 : 1st woman having a boy, B 2 : 2nd woman having a boy. P(B 1 B 2). G 1 : 1st woman having a girl, G 2 : 2nd woman having a girl. P(G 1 G 2). P(B 1 G 2) and P(G 1 B 2). A : At least one boy. P(B 1 B 2 A).

15 Sensitivity, and Specificity The probability of a positive test result given that the individual actually tested actually has the disease is called the sensitivity of a test. The probability of a negative test result given that the individual actually tested does not have the disease is called the specificity of a test.

16 Example Cervical cancer Pap Smear % of the tests performed on women with cancer resulted in false negative outcomes. A false negative occurs when the test of a woman who has cancer of the cervix incorrectly indicates she does not. P(test negative cancer) =

17 Example Cervical cancer Pap Smear % of the tests performed on women with cancer resulted in false negative outcomes. A false negative occurs when the test of a woman who has cancer of the cervix incorrectly indicates she does not. P(test negative cancer) = P(test negative cancer) = P(test positive cancer) = (sensitivity).

18 Example Cervical cancer Pap Smear % of the tests performed on women with cancer resulted in false negative outcomes. A false negative occurs when the test of a woman who has cancer of the cervix incorrectly indicates she does not. P(test negative cancer) = P(test negative cancer) = P(test positive cancer) = (sensitivity). P(test positive no cancer) =

19 Example Cervical cancer Pap Smear % of the tests performed on women with cancer resulted in false negative outcomes. A false negative occurs when the test of a woman who has cancer of the cervix incorrectly indicates she does not. P(test negative cancer) = P(test negative cancer) = P(test positive cancer) = (sensitivity). P(test positive no cancer) = P(test positive no cancer) = P(test negative no cancer) = (specificity).

20 Relative Risk Relative Risk (RR) is useful when we want to compare the probabilities of disease in two different groups. RR is the chance that a member of a group receiving some exposure will develop disease relative to the chance that a member of an unexposed group will develop the same disease: RR = P(disease exposed) P(disease unexposed)

21 Relative Risk Relative Risk (RR) is useful when we want to compare the probabilities of disease in two different groups. RR is the chance that a member of a group receiving some exposure will develop disease relative to the chance that a member of an unexposed group will develop the same disease: RR = P(disease exposed) P(disease unexposed) RR=1.0 indicates... RR>1.0 indicates... RR<1.0 indicates...

22 Example Breast Cancer A woman is exposed, if she first gave birth at the age of 25 or older. In a sample of 4540 women who gave birth to their first child before the age of 25, 65 developed breast cancer. Of the 1628 women who first gave birth at age 25 or older, 31 were diagnosed with breast cancer. RR of developing breast cancer is: