One of two biased coins A and B is selected and flipped 3 times. Let A be the event that coin A is selected and B be the event that coin B is selected, with probabilities p(A) = 0.4 and p(B) = 0.6. When coin A is flipped, the probability of heads is 0.2. When coin B is flipped, the probability of heads is 0.8. Let HHH be the event that the selected coin comes up heads 3 times. Complete the values X, Y, and Z in Bayes' Theorem to determine the probability coin B was chosen if all 3 flips come up heads. X Ex: 0.123-0.6 p(B|HHH) = Ex: 0.123 Y 0.6+Ex: 0.123 Ꮓ . 0.4
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- In a communication tower, let k be the number of active (non-silent) speakers in a group of six non-interacting (i.e., independent) speakers. Suppose that a speaker is active with probability 1/4. Compute the following 1. Using the appropriate mathematical formulas to find the probability that the number of active speakers is greater than three. 2. Using MATLAB to simulate the random experiment and validate the results evaluated in 1. 3. Plot the probability density function of the random variable k. (a) What is basic problem area? (b) Why is this topic important? (c) What ideas do you have for future directions for this work? (d) What did you learn from this project?The sample space of a random experiment is (a, b, c, d, e, f, and each outcome is equally likely. A random variable is defined as follows: outcome X с d TEISEER 1.5 1.5 a b 0 2 Determine the probability mass function of X. Use the proba- bility mass function to determine the following probabilities: (a) P(X= 1.5) (b) P(0.5 3) (d) P(0 ≤X<2) (e) P(X=0 or X = 2)The random variable X and Y are independent. The random variable X has the probability distribution P(X=0) = 0.4 and P(X=1) = 0.6. The random variable Y has the probability %3D distribution P(Y=0) = 0.2 and P(Y=3) = 0.8. Fill the following Joint distribution table: Y P(X,Y) 3 1 1 Note: round your answer to decimal places.
- 2. Given a Sample Space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, Event A = {1, 3, 4, 7, 9}, and Event B = {3, 7, 9, 11, 12, 13} Find the probability P(A|B). State your answer as a value with one digit after the decimal point.Consider a die in which the outcome 1 is twice as likely to occur as any other outcome. Let X be the random variable that is the outcome of rolling this die. Compute the expected value of X and variance of X.For the Covid-19 pandemic, let’s make the following assumptions: (i) the natural infection rate is 20% (i.e., a person who is not vaccinated has 20% chance of being infected); (ii) the vaccine is 90% effective at preventing Covid-19 infection (i.e., a vaccinated person has only 2% chance of being infected); (iii) the probability of a random person being vaccinated is 60%. Without knowing whether or not a person is vaccinated, what is the probability that this person will be infected? Given that a person is infected, what is the conditional probability that this person is vaccinated?
- If the probability of an event is 62%, what is the probability of its complement?5. The probability distribution of discrete random variable X is given by k for x = 1, 2, 3 x +1 P(X = x) %3DThis exercise asks you to (partly) verify that linearity of expectation holds even if the random variables are not independent. Consider two 6-sided dice that are rolled independently. What is the expected value of the sum? Now consider the case where the first die is rolled normally and then the second die is set equal to the value shown on the first die. What is the expected value of the sum? Now consider the case where the first die is rolled normally and the second die is set equal to 7 minus the value of the first die. What is the expected value of the sum?
- ints Assume that females have pulse rates that are normally distributed with a mean of u 72.0 beats per minute and a standard deviation of a 12.5 beats per minute. Complete parts (a) through (c). below. a. If 1 adult female is randomly selected, find the probability that her pulse rate is less than 79 beats per minute. The probability is (Round to four decimal places as needed.) b. If 4 adult females are randomly selected, find the probability that they have pulse rates with a mean less than 79 beats per minute. The probability is (Round to four decimal places as needed) c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 307 OA. Since the mean pulse rate exceeds 30, the distribution of sample means is a normal distribution for any sample size. OB. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size OC. Since the distribution is of individuals, not sample…Suppose that the length of a phone call in minutes is an exponential random variable with parameter λ=1/10. If someone arrives immediately ahead of you at a public telephone booth, find the probability that you will have to wait, (a) More than 10 minutes (b) Between 10 and 20 minutesX is a discrete random variable with the following PMF: P(X = 1.4) = 0.25 P(X = -9.7) = 0.25 P(X = 10) = 0.5 Find the standard deviation of X.