`A_1`, `A_2` and `A_3` are 3 events definds on a simple space s.
Use sets-theoric rotat6ions to represent the following events.
None of them will occur.

A_1 , A_2 and A_3 are 3 events definds on a simple space s. Use sets-theory rotations to represent the following events. Exactly one of them will occur.

A_1 , A_2 and A_3 are 3 events definds on a simple space s. Use sets-theoric rotations to represent the following events. At least two of them will occur.

If A, A_1,A_2,A_3 are the areas of the inscribed and escribed circles of a DeltaABC, then

Let A_r ,r=1,2,3, , be the points on the number line such that O A_1,O A_2,O A_3dot are in G P , where O is the origin, and the common ratio of the G P be a positive proper fraction. Let M , be the middle point of the line segment A_r A_(r+1.) Then the value of sum_(r=1)^ooO M_r is equal to (a) (O A_1(O S A_1-O A_2))/(2(O A_1+O A_2)) (b) (O A_1(O A_2+O A_1))/(2(O A_1-O A_2) (c) (O A_1)/(2(O A_1-O A_2)) (d) oo

In a four-dimensional space where unit vectors along the axes are hat i , hat j , hat k and hat l , and vec a_1, vec a_2, vec a_3, vec a_4 are four non-zero vectors such that no vector can be expressed as a linear combination of others and (lambda-1)( vec a_1- vec a_2)+mu( vec a_2+ vec a_3)+gamma( vec a_3+ vec a_4-2 vec a_2)+ vec a_3+delta vec a_4=0, then

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 P(A_1) = 0.2 , P(A_2) = 0.1 and P(A_3) = 0.3 and these events are mutually independent, then what is the value of P(A_1 nn A_2 nn A_3) ?

For three event A_1 , A_2 and A_3 state and prove Bonferroni's inequality.

If A_1 , A_2 , A_3 are mutually exclusive, mutually independent and exhaustive, then the probability that A_1 , A_2 , A_3 occur simultaneously is