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Probability.txt
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Probability.txt
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## Probability of Events
Question 1: Disjoint and Independent
Suppose A and B are events, each of which have some chance of occurring (i.e. P(A) and P(B) are both greater than 0). Suppose further that A and B are disjoint. Which of the following is true and why:(a) A and B can not be independent.(b) A and B must be independent.(c) A and B can be either independent or not independent.To consider which of these is true, it may be helpful to think of examples of disjoint events and consider their independence or lack thereof.
Question 2: Probability Distribution
Suppose that families are allowed to have only one son, but may continue having children until a son is born. (Suppose that all families do this.) Suppose further that the probability that a child born is a male is 1/2, and that the gender of children in a family are independent. Find the probability distribution for X = number of girls in the family. I will get you started by noting that the probability that X = 0 is 1/2, since this is the probability of the event {B}. So now you have to figure out the probability of events such as these [X = 1] = {GB}.Figure out the probability for X = 1 all the way to X = 9. Show your work as you find the probabilities (i.e. don’t just report the probabilities, give the calculations that you used to find them).Question 3: Another DistributionThe probability of a male birth is actually not quite 50%, but actually a little bit more. (Here is a link to an interesting article about thathttp://www.scientificamerican.com/article.cfm?id=is-a-pregnant-womans-chan )Suppose we wanted to redo the calculate above and use the probability 51.2% for a male birth.a. Describe how you could use this percentage and determine the distribution for X = number of girls when X = 1, …, 9. b. What effect do you think this would have on our results? Do you think the mean number of children per family and ratio of boys to girls would change? In what way? Why? (You don't have to actually carry out another simulation, but just support your answer with a reasonable explanation.)
Question 4: Independence and Correlation
In class we talked about how averaging your independent investments in a portfolio over two stocks could make the standard deviation of the return smaller than if you had put all your funds into one investment. That is, the standard deviation of a average of independent random variables with the same standard deviation is less than the the standard deviation of just one of the random variables.
What do you think would happen if the two were NOT independent, but rather were positively correlated? For example, suppose you split your funds between the stocks of two companies in the same industry, say consumer electronics (e.g., Dell and Apple). Do you think your return would be less variable, more variable, or the same variability as if the two were independent? Do you think your return would be less variable, more variable, or the same variability as if you put all your money in one of them (say Dell). Why?
Question 5: Bayes Theorem
Watch the TED talk here: http://www.ted.com/talks/lang/en/peter_donnelly_shows_how_stats_fool_juries.htmlNow consider the following variation on the problem Donnelly discusses. Suppose that the probability that an infant dies of crib death is 1/8500. But as discussed, crib deaths are related to environmental or genetic factors, so if one child in a family dies of crib death, subsequent children in the family are more likely to die of crib death. Specifically suppose the probability that a second infant in a family dies of crib death given that the first one died of crib death is increased to 1/100. Now, what is the probability that both children in a family die of crib death? Are crib deaths of children in a family independent?