How To Understand Probability Theory

Transcription

1 WEEK #21: Genetics, Probability Theory, Conditional Probability Goals: Textbook reading for Week #21: Study Adler Section 6.3, 6.4, 6.5.

2 2 Probability Theory Review From Section 6.3 Venn diagrams are a graphical way to represent sets, their overlap, and their relative sizes. In probability theory, each set represents an event. Write out an expression for the shaded region of each of the following diagrams in terms of sets. and then write out the probability of the shaded event. A B

3 Week 21 Genetics, Probability Theory, Conditional Probability 3

4 4 B A Question: If the sets A and B represent events, which of the events is the most likely, or has the highest probability? A. A B. B Question: Which of their complement events is more likely? A. A c B. B c

5 Week 21 Genetics, Probability Theory, Conditional Probability 5 A B Question: Which of the phrases below would accurately describe the events A and B? A. mutually exclusive B. collectively exhaustive Question: How would you compute the probability of the event A B? A. Pr(A) Pr(B) B. Pr(A) + Pr(B) C. 1 (Pr(A) + Pr(B))

6 6 B A Question: The probability of the event A B is A. 1 B. Pr(A) + Pr(B) C. Pr(A) Pr(B) D. 0

7 Week 21 Genetics, Probability Theory, Conditional Probability 7 Conditional Probability From Section 6.4 Working with our new ideas of events and sample spaces, we can now more easily define and compute probabilities for more interesting cases. Example: Consider a genetics example studying the height of plants, with the possible alleles b and B. Suppose that the B allele is dominant, and indicates tall plant height. If the plant population has the standard 25%/50%/25% bb/bb/bb genotype probabilities, what is the probability that a random plant is tall?

8 8 We now add a condition on our reasoning. If we select a tall plant, what is the probability that it is Bb heterozygous? Argue first from intuition, or using a Venn diagram.

9 Week 21 Genetics, Probability Theory, Conditional Probability 9 Law of Conditional Probability Pr(A B) = for any event B for which Pr(B) 0, where Pr(A B) Pr(B) Pr(A B)= the probability of A occurring given that B occurred/is true An alternative phrasing for our previous question is What is the probability of a plant being heterozygous given that it is tall?

10 10 Define the appropriate single events, and then use the Law of Conditional Probability to compute the probability that a plant is heterozygous, given that it is tall.

11 Week 21 Genetics, Probability Theory, Conditional Probability 11 There are some interesting sub-cases for conditional probability. Compare the probabilities that a tall plant is BB homozygous, and that a BB homozygous plant is tall. Sketch a Venn diagram for these scenarios. More examples of conditional probabilities for subset and disjoint event relationships are shown on page 516.

12 12 Law of Total Probability For an event A, and a set of collectively exhaustive and mutually exclusive events E i, Pr(A) = Pr(A E i ) Pr(E i ) Draw a Venn Diagram to visualize this Law, using 3 events E 1, E 2, E 3, and a distinct event A.

13 Week 21 Genetics, Probability Theory, Conditional Probability 13 The Law of Total Probability can be helpful in doing quick calculations based on disease statistics. In Canada, 20% of the population is 60 years or older. The chance of a new occurrence of cancer in people 60 years or older is 2% per year, but only 0.15% per year in younger people. Compute the probability of a random citizen developing cancer this year.

14 14 Bayes Theorem For any two events A and B Pr(B A) = Pr(A B)Pr(B) Pr(A) Derive this law from the formula for conditional probability.

15 Week 21 Genetics, Probability Theory, Conditional Probability 15 Bayes Theorem has a wide variety of applications. One of the most famous is in explaining why we do not do population-wide screening for rare diseases. A rare but serious disease occurs in 2% of the population. Doctors develop a test that registers positive results for positive cases 99% of the time, and registers positive results for negative cases only 3% of the time. Would you describe this test as generally accurate?

16 16 Compute the probability that you, walking into doctor s office today, would test positive for the disease, using the test described earlier.

17 Week 21 Genetics, Probability Theory, Conditional Probability 17 Assume you do test positive: how concerned should you be? Compute the appropriate probability.

18 18 Show how pre-screening can ameliorate this disease screening problem. Say that having high-blood pressure increases the likelihood of the disease to 25%. (25% of those with high blood pressure exhibit this disease.) If you present to your doctor with high-blood pressure, what is the probability now that a positive test indicates the presence of the disease?

19 Week 21 Genetics, Probability Theory, Conditional Probability 19 Here is a more light-hearted example. Example: In the recent book Outliers, Malcolm Gladwell makes note of an interesting fact: birthdays in the population are uniformly distributed (25% per quarter), but among NHL players, 31% have birthdays in the first quarter (Jan-Mar). In Ontario, roughly 20% of the draft-eligible minor leaguers eventually play in the NHL, and the minor league players have a roughly uniform birth date distribution, just like the whole population.

20 20 If you are draft-eligible this year, and born in between January and March, what is the probability that you will play in the NHL?

21 Week 21 Genetics, Probability Theory, Conditional Probability 21 Independence of Events From Section 6.5 Consider the classical probability exercise of flipping a coin twice. Write out the possible set of outcomes i.e. the sample space.

22 22 What is Pr(HT) (heads first, followed by a tail)? Is the probability of HT the same as that of any of the other outcomes? Does the outcome of the first coin toss change the probabilities of the second coin toss? i.e. if you knew the first coin came up heads, would you change a bet on the second flip? Events like repeated coin flips are considered to be independent, because knowing whether one event occurred (or not) gives you no additional information about the probability of the other events.

23 Week 21 Genetics, Probability Theory, Conditional Probability 23 Some other examples of independent events: Examples of non-independent events:

24 24 The concept of independence is a very intuitive idea, but what does it mean mathematically? Events A and B are independent if Pr(A B) = Pr(A). It is irrelevant that event B occurred since it does not have any influence (or change any bets we might make) on A occurring. Also notice that if events A and B are independent, then both and Pr(A B) = Pr(A) Verify this with Bayes Theorem. Pr(B A) = Pr(B)

25 Week 21 Genetics, Probability Theory, Conditional Probability 25 As a consequence of the independence relation, what can we write as an equation for Pr(A B) if A and B are independent? The idea of independence can be extended to more than two events. For example, events A, B, and C are mutually independent if Pr(A B C) = Pr(A)Pr(B)Pr(C).

26 26 Example: Consider a group of five students in MATH 122. For now, we will assume that the students do not know each other and that the probability of any one of them coming to class is 5/6 (perhaps the average student misses a Monday class once every two weeks). What is the probability that all five students attend class?

27 Week 21 Genetics, Probability Theory, Conditional Probability 27 Now, suppose that within the group of five students there are two roommates. Student A comes to class 5/6 of the time, like most students, but also makes an effort to wake and bring along his roommate when he goes to class. This means that student B will be in class with probability 0.9 if student A goes, but only 0.2 if student A does not. With this new situation, what is the probability that all five students attend class?

28 28 In the second scenario, what is the probability that none of the five students attend class?

29 Week 21 Genetics, Probability Theory, Conditional Probability 29 Example: It s a Friday night in the jungle, and the cockatoos are on the prowl. A particular female cockatoo finds that the males can be separated into three categories: ones that are have a particularly sweet mating call, ones that have attractive plumage, and, well, the others. The population consists of: 25% good singers, 15% well-groomed, and 60% of the rest. The female prefers them in that order (vocal, well-groomed, and then average). The female will pair up with the best mate she can find. Suppose the female cockatoo decides to pair with her best option after meeting two male cockatoos. What is the probability that she has to settle for an average mate?

30 30 What did we assume about independence in the above calculation?