Now,
and
that implies
Here,
So, A and B are independent.
Similarly,
that implies
Here,
So, A and C are independent.
and also
that implies
Here,
So, B and C are also independent.
Hence, A, B and C are mutually independent.
l. An experiment has four equiprobable outcomes, 'S-{ 1 ,2, 3, 4). We define three events:...
1 . An experiment has four equiprobable outcomes. S { 2.3) and C { 1 , 2.3.4). We define three events A { 1 . 2 }. В { 1.3). Are A. B and C independent? Justify your answer.
Problems to be turned in: 1. An experiment has four equiprobable outcomes, S (1,2,3,4). We define three events: A 1,2). B (2,3) and C-(,3). Are A, B and C independent? Justify your answer.
(b) Construct an experiment and three associated events A, B and C such that A and B are not independent, but AC and BC are independent. Justify your answer with calculations
In an experiment, there are n independent trials. For each trial, there are three outcomes, A, B, and C. For each trial, the probability of outcome A is 0.20; the probability of outcome B is 0.70; and the probability of outcome C is 0.10. Suppose there are 10 trials. (a) Can we use the binomial experiment model to determine the probability of four outcomes of type A, five of type B, and one of type C? Explain. No. A binomial...
In an experiment, there are n independent trials. For each trial, there are three outcomes, A, B, and C. For each trial, the probability of outcome A is 0.40; the probability of outcome B is 0.40; and the probability of outcome C is 0.20. Suppose there are 10 trials. Can we use the binomial experiment model to determine the probability of four outcomes of type A, five of type B, and one of type C? Explain.
Problem 1. A biased coin with probability plandin with a Heads is lipped 4 times. (a) Define the basic random variables and give the sample space and assign probabilities to the outcomes. (b) Let X be the total number of Heads in the four flips Draw a Venn diagrain showing the five events X = ii 0,1,2,3,4 as well as the sample space and the outcomes. Is X a random variable? c) Are the events X 1 and X 2...
(i) Suppose L = {(1, 4, 2, 2),(2, 2, 1, 2),(2, 4, 2, 1)}. Is L linearly independent in R^4 ? Justify your answer. (ii) Suppose S = { 0 0 1 0 , 0 2 3 0 , 4 1 0 0 }. Is S linearly independent in M2(R)? Is span(S) = M2(R)? Justfy your answers.
Question 4: You roll a fair die once. Define the events A = “the result is one of the numbers 1, 3, and 4”, B = “the result is one of the numbers 3, 4, 5, and 6”. • Are the events A and B independent? Jusitfy your answer using the definition of independence. • Are the events A and B independent? Jusitfy your answer using the definition of conditional probability. Question 4: You roll a fair die once. Define...
A random experiment can result in the outcomes 1, 2, 3, 4, 5, a, b, c. Let event M contain the outcomes a,b, c and let event N contain the outcomes 1, 2, 3, a. Then, which of the following is true? a. b. {b, c} c. {1, 2, 3, 4, 5, a, b ,c} d.
7. Suppose that an experiment has two outcomes 0 or 1 (such as flipping a coin). Suppose that independent experiments and for the ith experiment you let the random variable X Ber(p) with we will assume for this problem that p is the same for each i). Then, you run n tell you the outcome for 1 isn. Then we can assume that for each i, that X p P(X 1) (where ΣΧ. let X (a) What is the state...