Question

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Exercice 1 If A and B are not mutually exclusive events, then the number of times A occurs in N trials can be written as NA = NAB + Nage, where Nag stands for the number of times A occurs without B. Use this notation to show that P(ABC) = P(A) - P(AB). ............. Exercice 2 A discrete RV has two possible values, a and b. Find the mean, second moment, and standard deviation in terms of p = P(X = a).

Answer 1

For Question 1, you can understand the concept through Venn- Diagram as well as through the given equation. Now, it is given that

Method-1

N_A=N_AB+ N_AB^c

N_A - N_AB = N_AB^c ............(1)

Also, Probability for AB^c, P(AB^c) = N_AB^c / N, where N are the total number of trials.

Also, from dividng equation 1 by N we get

N_A / N - N_AB/ N = N_AB^c /N

P(A) - P(AB) = P(AB^c)

Hence, proved !

Method-2 By Venn Diagram

Now, AB^c refers to the region which is obtained by A and excluding B. Thus, it can be shown as,

Now, this region can also be referred as $A\bigcap B$ or AB removed from A. Thus A - $A\bigcap B$ .

Therefore, P(A) - P(AB) = P(AB^c)