Conditional probability, and probability trees
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Researchers randomly assigned 72 chronic cocaine users to receive either an antidepressant, lithium, or a placebo. 48 of the 72 users relapsed. Of those who received the antidepressant, 10 of 24 relapsed, giving a 42% probability of relapse among those receiving the antidepressant. The probability of relapse was higher for those receiving lithium (75%) or placebo (83%). Knowing a patient relapsed, the probability they received the antidepressant was 21%.
Conditional probability, and probability trees
- 1. + Conditional Probability, and Probability Trees, Slides edited by Valerio Di Fonzo for www.globalpolis.org Based on the work of Mine Çetinkaya-Rundel of OpenIntro The slides may be copied, edited, and/or shared via the CC BY-SA license Some images may be included under fair use guidelines (educational purposes)
- 2. Relapse Researchers randomly assigned 72 chronic users of cocaine into three groups: desipramine (antidepressant), lithium (standard treatment for cocaine) and placebo. Results of the study are summarized below. http://www.oswego.edu/~srp/stats/2_way_tbl_1.htm
- 3. Marginal probability What is the probability that a patient relapsed? P(relapsed) = 48 / 72 ~ 0.67
- 4. Joint probability What is the probability that a patient received the antidepressant (desipramine) and relapsed? P(relapsed and desipramine) = 10 / 72 ~ 0.14
- 5. Conditional probability The conditional probability of the outcome of interest A given condition B is calculated as
- 6. Conditional probability (cont.) If we know that a patient received the antidepressant (desipramine), what is the probability that they relapsed? P(relapse | desipramine) = 10 / 24 ~ 0.42
- 7. Conditional probability (cont.) If we know that a patient received the antidepressant (desipramine), what is the probability that they relapsed? P(relapse | desipramine) = 10 / 24 ~ 0.42 P(relapse | lithium) = 18 / 24 ~ 0.75 P(relapse | placebo) = 20 / 24 ~ 0.83
- 8. Conditional probability (cont.) If we know that a patient relapsed, what is the probability that they received the antidepressant (desipramine)? P(desipramine | relapse) = 10 / 48 ~ 0.21 P(lithium | relapse) = 18 / 48 ~ 0.38 P(placebo | relapse) = 20 / 48 ~ 0.42
- 9. General multiplication rule ● Earlier we saw that if two events are independent, their joint probability is simply the product of their probabilities. If the events are not believed to be independent, the joint probability is calculated slightly differently. ● If A and B represent two outcomes or events, then P(A and B) = P(A | B) x P(B) ● Note that this formula is simply the conditional probability formula, rearranged. ● It is useful to think of A as the outcome of interest and B as the condition.
- 10. Independence and conditional probabilities Consider the following (hypothetical) distribution of gender and major of students in an introductory statistics class: ● The probability that a randomly selected student is a social science major is 60 / 100 = 0.6. ● The probability that a randomly selected student is a social science major given that they are female is 30 / 50 = 0.6. ● Since P(SS | M) also equals 0.6, major of students in this class does not depend on their gender: P(SS | F) = P(SS).
- 11. Independence and conditional probabilities (cont.) Generically, if P(A | B) = P(A) then the events A and B are said to be independent. ● Conceptually: Giving B doesn’t tell us anything about A. ● Mathematically: We know that if events A and B are independent, P(A and B) = P(A) x P(B). Then,
- 14. If an individual from Swaziland has tested positive, what is the probability that he carries HIV? P(HIV | +) = 0.82 There is an 82% chance that an individual from Swaziland who has tested positive actually carries HIV.