Consider the probability space where $=[0,1]\backslash$times $[0,1],$ F $=\backslash$s Consider the probability space where $=[0,1][0,1],F=(\{[a,b][c,d]$:$a,b,c,$din$[0,1]\}),$

and the product measure $P$ with the property that $P[[a,b][c,d]]=(b-a)(d-c)$ whenever

$ab,cd.$

Consider the following subset of $$:$E=\{({}_1,{}_2)in$:${}_{1}in[0,1],{}_2={}_1\}.$

$($a$)$ Why is $E$ an event $($i$.$e$.,$ EinF $)?$

$($b$)$ What is the value of $P[E]?$igma $(\{[$a$,$b$]\backslash$times $[$c$,$d$]$ : a$,$b$,$ c$,$d in $[0,1]\}),$

and the product measure P with the property that P$[[$a$,$b$]\backslash$times $[$c$,$d$]]=($b$$a$)($d $$c$)$ whenever

a b$,$ c d$.$

Consider the following subset of $$: E $=\{(\backslash$omega $1,\backslash$omega $2)$ in $$ : $\backslash$omega $1$ in $[0,1],\backslash$omega $2=\backslash$omega $1\}.$

$($a$)$ Why is E an event $($i$.$e$.,$ E in F$)?$

$($b$)$ What is the value of P$[$E$]?$