Lab 4b for Math 17 Probability Practice

Lab 4b for Math 17: Probability Practice 1 The Law of Total Probability and More The law of total probability gives us some additional options to find probabilities of events. This was actually how we found the denominator probability in the Baye’s Rule Formula. For an event A, and set of events Bi , i = 1, . . . , n, which partition the entire sample space, you can find the probability of A using the probabilities of Bi and the conditional probability of A given each Bi . That is, the law of total probability states that: P (A) = n X P (Bi )P (A|Bi ). i=1 For an interesting application of the law of total probability, consider the following situation. As a researcher, you need to ask a very sensitive question of your subjects (yes/no) and you aren’t sure they are going to reply honestly. Warner’s Randomized Response Model was designed for this situation based on the law of total probability. Before we implement the model, let’s decide on a not-so-sensitive question that you all want to learn the answer to. For example, the probability that students in the class have had an internship while at Amherst. 5. Our question of interest (yes/no) is As the researcher, you would give the following directions to your subjects as well as a coin. Flip the coin twice, and record the flips just for yourself. If the first flip was heads, answer question 1. If the first flip was tails, answer question 2. On the slip of paper, record ONLY the yes/no answer to your question (not the flips). Additionally, you would need the following questions set up: Question 1 (Q1). (Question of Interest) Question 2 (Q2). Was the second flip heads? Thinking about the setup and the fact that we are interested in the probability of a yes reply to question one, we could use the law of total probability as follows: P (Y es) = P (Y es and Q1) + P (Y es and Q2) = P (Y es|Q1)P (Q1) + P (Y es|Q2)P (Q2). Note that we can fill in values for most of these probabilities already. P (Y es) = P (Y es|Q1)(.5) + (.5)(.5) Compute P(Yes) based on the class responses and then compute P (Y es|Q1). 1 2 Basic Probability Practice - Python Eggs The following table contains data on number of eggs hatched and not for 3 different temperature settings for python eggs, as recorded by scientists studying environmental impact of temperature changes on python reproduction. Temp/Hatch? Cold Neutral Hot Total Yes 16 38 75 129 No 11 18 29 58 Total 27 56 104 187 1. Name an event related to this scenario and its complement. 2. What is the probability a randomly selected egg from this experiment hatched? 3. Given each temperature setting (do a calculation for cold, neutral, and hot), what is the probability a randomly selected egg from that level hatched? 4. The calculations in 2 and 3 allow you to check for independence (be sure you understand why). Are the events temperature and hatch status independent? 5. Use the probabilities from 2 and 3 as population probabilities to complete the following: a. What is the probability a randomly selected python who leaves her egg at a neutral temperature will have a hatched egg? b. Suppose two randomly selected pythons leave their eggs at cold temperatures. What is the probability that both eggs hatch? c. Suppose a randomly selected python leaves her egg at an unknown temperature. What is the probability that her egg doesn’t hatch? d. Suppose three randomly selected pythons leave their eggs at hot temperatures. What is the probability that all three eggs don’t hatch? e. In the same setting as d., what is the probability at least one of the three eggs hatches? 2 3 Practice with Bayes, Trees (Diagrams), or Hypothetical 1000 Many species of fish are susceptible to high levels of nitrate in their water, which can in turn cause Brown Blood disease, where the fish literally suffocate to death because their blood can no longer carry oxygen. During the capture stage of a capture-recapture sampling problem, you decide to take samples of fish blood to test for Brown Blood disease (assume you have developed a test for this). Assume that in the region where you are taking samples of fish, 5 percent of sampling locations (ponds, lakes, rivers, etc.) have high nitrite concentrations and the fish are suffering from Brown Blood disease. The test you have developed will detect Brown Blood disease 90 percent of the time when the fish actually has it, but also indicates the fish has the disease 10 percent of the time when the fish really does not have the disease. Use probability rules, a tree diagram, or make a table of a hypothetical 1000 fish to assist you in determining the following probabilities: 1. The probability a randomly selected fish will test positive 2. The probability a randomly selected fish actually has the disease given that they test positive 3. The probability a randomly selected fish has the disease and it was not detected 3 4 Random Variable Practice Suppose you and a group of friends (say, you and 5 friends, so 6 people total) plan to attend a movie showing this weekend. Let X be the number of people in your group (in addition to you) that show up on time to travel to the movie showing. Suppose the partial probability distribution of X is given as follows: x P (X = x) 0 .05 1 .10 2 .15 3 .2 4 .3 5 a. What is P (X = 5)? b. What is the expected value of X? c. What is the variance of X? d. Suppose tickets cost 9 dollars, and you are treating everyone who shows up on time to the movie. What is the expected cost to you for the trip? (Note you do have to pay for yourself too!). e. Now suppose you decide to declare a movie showing “successful” if at least 3 of your friends go with you. What is P(successful showing)? f. Suppose you decide to go to a showing each weekend for the next 4 weekends. Let Y be the number of successful showings out of the 4 showings. If you make some assumptions, the distribution of Y could be binomial. What binomial distribution would Y have (give the 2 parameters), and what assumptions would you have to make? g. Using the model from f, what is P (Y > 2)? 4 5 Probability Question to Turn In A study investigating the relationship between coronary heart disease and anger levels (measured using the Spielberger Trait Anger Scale) resulted in the following table classifying the 8474 subjects based on heart disease status and anger level. HD/Anger Yes (CHD) No (CHD) Total Low 53 3057 3110 Moderate 110 4621 4731 High 27 606 633 Total 190 8284 8474 a. What is the probability a randomly selected subject had coronary heart disease? Does this probability represent a sample proportion or population proportion? b. What is the probability a randomly selected subject had high anger levels? c. What is the probability a randomly selected subject had coronary heart disease if it was known the subject had high anger levels? d. Can you conclude that heart disease status and anger levels are independent? Explain briefly. For the last 2 parts, treat the probabilities as population probabilities (i.e. do not worry about the effect of sampling without replacement). e. What is the probability that 2 randomly selected subjects will have no CHD given that each of them has a high anger level? f. What is the probability that a random sample of 2 subjects will yield one subject with CHD and moderate anger level and another subject with no CHD and moderate anger level? 5