Question

Explain when will two events be independent and when will two events be mutually exclusive. Can two mutually exclusive events be independent also? Can two independent events be mutually exclusive? Suppose the experiment is roll two dice. Consider events E= both numbers are even. F = both numbers are odd, Are E and F mutually exclusive? Are they independent?  

Consider events U and V. U= the first number is even, V= the second number is even.  Are U and V mutually exclusive? Are they independent?  

Answer 1

TuD events X amd y ase said to be indopendont PCXIY) P(X) if , PCx)>o e PCxoy) PCy) PC)o P(X) ahre P(XIy) s the corditional protability of X gicen y Tuo events X+ Y are said to be mutually exclusive ik intersection of both eventsis empty ies PCXOY)=O mutually excluive events cam not be indaterdot. As, Muitually exclwiue events can not occur at the same time. Xoccuss then y com not occusr Turn and vice-versa. Unles the evert is imbssile) Tuo indshinda-t evets cam not be mtual erclusve 6 X4Yare indahindat the P(xoy)-f(P P(x) > o and Ply) 4 svich that tX4 y are mutually excusise PCny eithu P(x) -o e Ply)o, which is a centradiction as P(x)>o t P(y) > o. kene, two indehendant eoents can not be mutuale esclusie

tun dice E= bath noS are even =s(2,2), C2.9), (2,6), (46),u4), 6,6) 2) (6,2) (6,4) F= Roth nas are odd (11), (,1(5), (5,) (3.3) (3,5) Hare, EnF PCENF)- exclusive evonts Hena Xyare net indsperdat nt indobendat re, EFax Ufirst no is evn (G1) (621(63) (64), G ),69 CL2) (22) (321(4,+) (62) (14), (2,4) (3,4)2 64),61) (101 (29, (361, (4.6), (s66 Hare Unv nat mutually eaclusive UnV-e),,9, U2),426), (G2),(64)66)3 PCU)= -P(v) I! V = 9 P(onV)= 36 4 3 36 0fV are indebendsnt fluov): P(U) PlU