# Let the sequence of events E1, E2, . . . , En be independent, and assume that P(Ei) = 1/(i+1). Show that P(E1 ∪ · · · ∪ En) = n/(n+1)

Question
Let the sequence of events E1, E2, . . . , En be independent, and assume that $$\displaystyle{P}{\left({E}{i}\right)}=\frac{{1}}{{{i}+{1}}}$$. Show that $$\displaystyle{P}{\left({E}{1}∪···∪{E}{n}\right)}=\frac{{n}}{{{n}+{1}}}$$

2021-02-13
Given: E1,E2...En, are independent events $$\displaystyle{P}{\left({E}{i}\right)}=\frac{{1}}{{{i}+{1}}}$$
To proof: $$\displaystyle{P}{\left({E}{1}{U}\ldots{U}{E}{n}\right)}=\frac{{n}}{{{n}+{1}}}$$
Use the Complement rule: $$\displaystyle{P}{\left({A}^{{c}}\right)}={P}{\left(\neg{A}\right)}={1}-{P}{\left({A}\right)}$$
PSKP((Ei)^c)=1-P(Ei) =1-(1/(i+1)) =(i+1)/(i+1)-(1/(i+1)) =i/(i+1)ZSK
Since the events E1, E2..., En are independent, the events $$\displaystyle{\left({E}{1}\right)}^{{c}},{\left({E}{2}\right)}^{{c}}\ldots{\left({E}{n}\right)}^{{c}}$$ are alslo independent.
We can use the multification rule for independent events PSKP(A ∪ B)= P(A and B) = P(A)xP(B)ZSK
PSKP((E1)^c)⋂...⋂((En)^c)=P(E1)^c*...*P((En)^c) =1/(1+1)*(2/(2+1))*(3/(3+1))*...*(n/(n+1) =1/2*2/3*3/4*...*n/(n+1) =(1(2)(3)(4)...(n))/(2(3)(4)...(n+1)) =1/(n+1)ZSK
Use the complement rule: $$\displaystyle{P}{\left({A}^{{c}}\right)}={P}{\left(\neg{A}\right)}={1}-{P}{\left({A}\right)}$$
PSKP(E1∪...∪En)=P(((E1)^c)⋂...⋂(En)^c)) =1-P(((E1)^c)⋂...⋂(En)^c)=1-(1/(n+1))=n/(n+1)ZSK

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The problem reads: Suppose $$\displaystyle{P}{\left({X}_{{1}}\right)}={.75}$$ and $$\displaystyle{P}{\left({Y}_{{2}}{\mid}{X}_{{1}}\right)}={.40}$$. What is the joint probability of $$\displaystyle{X}_{{1}}$$ and $$\displaystyle{Y}_{{2}}$$?
This is how I answered it. P($$\displaystyle{X}_{{1}}$$ and $$\displaystyle{Y}_{{2}}$$) $$\displaystyle={P}{\left({X}_{{1}}\right)}\times{P}{\left({Y}_{{1}}{\mid}{X}_{{1}}\right)}={.75}\times{.40}={0.3}.$$
What I don't understand is how do you get the $$\displaystyle{P}{\left({Y}_{{1}}{\mid}{X}_{{1}}\right)}$$? I am totally new to Statistices and I need to understand each part of the process in order to get the whole concept. Can anyone help me to understand why the P and X exist and what they represent?
$$S={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.$$