problem 1 suppose that p is a probability space consisting of four points which we label a b c d

Step 1: To show that E and G are independent, we need to show that P(E ∩ G) = P(E) * P(G).

Problem 1 suppose that p is a probability space consisting...

Problem 1. Suppose that (Ω, P) is a probability space consisting of four points, which we label a, b, c, d. (The one-element sets {a}, {b}, {c}, and {d} are the "simple" events of Ω). Suppose that P gives equal probability to each of these events. Thus P({a}) = P({b}) = P({c}) = P({d}) = 1/4. Consider the events E = {a, b}, F = {a, c}, and G = {a, d}. Show that E and G are independent, that F and G are independent, but E ∩ F is not independent of G.

Transcript

00:01 Hi, in this question we need to prove the given statement or if it is false we need to use the counter example to prove that.

00:09 So, here first statement is probability of a union b equals probability of a plus probability of b which is not true.

00:27 The correct statement is probability of a union b equals probability of a plus probability of b minus probability of a intersection b.

00:39 So, here we have to use the counter example.

00:44 Consider probability of c equals 1 by 10 and probability of g as 1 by 10 and probability of c intersection g equals 1 by 150.

00:58 So, we need to find the probability of c union g which is equal to 1 by 10 plus 1 by 10 minus 1 by 150.

01:12 So, which is equal to 9 by 150.

01:15 So, which is not equal to 2 by 10.

01:20 Hence conclude that the first statement is false.

01:27 Next move on to second statement.

01:29 In here we need to prove probability of a union b equals probability of a plus probability of b when a and b are mutually disjoined.

01:54 So, here probability of a intersection b by using the above proof we can consider probability of a union b equals probability of a plus probability of b minus probability of a intersection b.

02:12 If a and b are mutually disjoined then probability of a intersection b equals 0.

02:18 So, we can conclude that probability of a plus probability of b minus 0 which is equal to probability of a plus probability of b...

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