`A_1, A_2……….A_n` are n independent events such that `P(A_i) = 1 - q_i, I = 1, 2, 3…………….n` Prove that `P[overset(n)underset(i)uuu A_i] = 1 - q_1q_2……….q_n`

A_(1), A_(2), …, A_(n) are independent events such that P(A_(i))=1-q_(i), i=1,2, …, n . Prove that, P(A_(1) cup A_(2) cup … cup A_(n))=1-q_(1)q_(2)…q_(n) .

If the events E_1, E_2,………….E_n are independent and such that P(E_i^C) = i/(i+1) , i=1, 2,……n, then find the probability that at least one of the n events occur.

If E_(1) ,E_(2),E_(3)…E_(n) be n independent events such that P(Ei)=(1)/(1+i) for 1 le I le n then the chance that none of E_(1),E_(2),E_(3) … E_(n) occur is

If x_1 , x_2 ,……. x_n are real quantities. Prove that 1/n overset(n)underset(i=1)sum x i^2 ge (1/n overset(n)underset(i=1)sum|x i|)^2

Prove that 1/n overset(n)underset(E_1)sum |x_i - A| attains, minimum when A = Median.

Suppose A_1,A_2….. A_(30) are thirty sets each having 5 elements and B_1B_2…..B_n are n sets each having 3 elements ,Let overset(30)underset(i=1)bigcupA_1=overset(n)underset(j=1)bigcupB_j=s and each element of S belongs to exactly 10 of the A_1 and exactly 9 of the value of n.

If A_1,A_2, A_3,.. are sets such that n(A_i)=101-i , A_1 sup A_2 sup A_3 sup ...supA_100 and A=nn_(i=5)^100 A_i then n(A) is equal to

Suppose A_1 , A_2 ..... A_30 are thirty sets each with five elements and B_1,B_2....B_n are n sets each with three elements such that cap_(i=1)^30 A_i = cap_(j=1)^n B_j = S . If each elements of S belongs to exactly ten of the A_i s and exactly nine of the B_j's then the value of n is

An urn contains 4 tickets having the nos. 112, 121, 211, 222. A ticket is drawn at random. The even A_i = the event that the i^(th) digit is 1 (I = 1 to 3). Shwo that the events A_1 , A_2 , A_3 are pair wise independent but not mutually independent.

i^(2)=-1,"then"underset(n=0)overset(225)sumi^(n) is -