1. The weather predicts that the probability of precipitation is 30% for each of
ID: 3271047 • Letter: 1
Question
1. The weather predicts that the probability of precipitation is 30% for each of the next five days (Wednesday through Sunday) . Assume the fact that it rains on one day is independent of whether it rains on another day. The validity of this assumption could be argued.
a. What is the probability that your area receives rain all five days?
b. What is the probability they your area does not receive rain in the next five days?
c. What is the probability that it will rain on the fourth day (Saturday)?
d. What is the probability that it will rain on Saturday and Sunday, but does not rain on Wednesday, Thursday and Friday?
2. You have a developed a novel way of inserting genes from one species to another species. The success rate is 30% where success is measured by producing an organism where the gene is successfully passed from one generation to the next. How many trials would you have to perform to be 90% confident that at least one of the trials resulted in a success?
3. You are performing an experiment where animals (or plants---you choose) are exposed to a potential disease causing organism. If the probability of getting the disease is 0.4 during the duration of your study and you exposed 40 animals to the organism, how many do you expect to get the disease? What margin of error (+/- one standard deviation) would you attach to your estimate? What is the probability that five or fewer animals get the disease?
Hint: The expected number would be the mean. This is a binomial (have the disease or do not have the disease). Mean of a binomial is np, where n is the number of observations and p is the probability of success (getting the disease in this case). Variance of a binomial is np(1-p)
4. About 1 out of 8 invasive breast cancers are found in women younger than 45. Macon county had 12 cases of invasive breast cancers between 2005 and 2009. What is the probability that at least 2 cases were women younger than 45?
5. Bortkiewicz (1898) used the Poisson distribution to explain the chance of a Prussian cavalryman being killed by a horse kick. Ten army corps were observed over 20 years resulting in 200 observations (corp-years). If our data follow a Poisson process, we would anticipate that the expected number of deaths for each category would closely match the actual number of deaths that did occur. Please fill out the table and answer the questions that follow the table. Number of deaths per corp-year Number of corp-years Total number of deaths Probability Expected number of deaths 0 109 0 1 65 65 2 22 44 3 3 9 4 1 4 >4 0 0 Total 200 Formulas for mean and variance when only frequency data are presented.
a. Do you consider the actual and expected deaths are closely matched?
b. Is the average (mean) number of expected deaths close to the mean of the actual deaths?
c. Is the variance of the number of expected deaths close to the variance of the actual deaths?
6. Assume the probability that an entering college student will graduate is 0.4. Determine the probability that out of five students (a) none, (b) one, (c) at least one will graduate. a. Pr(none will graduate)= b. P(one will graduate)= c. P(at least one will graduate)= 7. If a new injection sterilizes male dogs 20% of the time, determine the probability that out of four fertile, randomly chosen dogs (a) 1, (b) 0, and (c) at most 2 dogs will be sterile after receiving the injection. a. P(1 sterile)= b. P(0 sterile)= c. P(at most 2 sterile)=
8. In one pen you have 2 Kiko’s and 4 Boer’s. In another pen you have 4 Kiko’s and 3 Boer’s. If you randomly select a goat to be slaughtered, what is the probability that it would be a Kiko. a. Your randomization procedure requires that the pen is randomly selected first and then a random goat within the chosen pen is taken. P(Kiko)=
b. Your randomization procedure randomly selects among the 13 goats without regard to pen. P(Kiko)=
9. If the probability that an individual suffers a bad reaction to a specific vaccination is 0.001, determine the probability that out of 2000 individuals more than 2 individuals suffer a bad reaction. Hint: Binomial probability problem, but 2000 is a large number and we do not want to calculate 2000 factorial. We can use the Poisson distribution to obtain an approximate value with less work, if we can obtain a value for . We substitute the mean of the binomial for . Therefore, =np=(2000)(0.001)=2. The below shows what you would get using the approximation compared to the binomial. P(2 using Poisson approximation)= 0.27067057 P(2 using Binomial)= 0.27080599 The above answers the question of the probability of exactly two bad reactions. The question to answer is the probability of more than 2 individuals.
Explanation / Answer
Q1) (i) The probability that your area receives rain all five days = (0.30^5) = 0.00243
(ii) The probability they your area does not receive rain in the next five days = (0.70^5) = 0.16807
(iii) the probability that it will rain on the fourth day (Saturday) = 0.30
(iv) the probability that it will rain on Saturday and Sunday, but does not rain on Wednesday, Thursday and Friday = (0.30*0.30*0.70*0.70*0.70) = 0.03087
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