Expectation of Random Variable

Question

Consider a quiz where a person is given two questions and he must decide which question to answer first. Question 1 will be answered correctly with probability of 0.8 and the person will then receive as prize \$100, while question 2 will be answered correctly with probability 0.5, and the person will then receive as prize $200.

If the first question is answered incorrectly then the quiz terminates and that person is not allowed to attempt the second question.

What is the expected amount of money that can he win?

 

My solution 

X – Random variable which denotes the amount of money won

X	P(X=x)
0$	0.5 +0.2 = 0.7
100$	0.8*0.5 = 0.4
200$	0.5*0.2 = 0.1
300$	0.8

 

P(X=0) = Either we answer Question 1 incorrectly or we answer Question 2 incorrectly

P(X=100) = We answer Question 1 correctly and answer Question 2 incorrectly

P(X=200) = We answer Question 2 correctly and answer Question 1 incorrectly

P(X=300) = We answer Question 1 and then Question 2 correctly in that order or

                    We answer Question 2 and then Question 1 correctly in that order

 

But if we sum P(X=0) + P(X=100) +  P(X= 200) + P(X=300) = 2

Where I am getting wrong?

Comments

amount                                     Probability

0 $ $ ---→ both are wrong → $0.2*0.5=0.1$

100---------->first one is correct and second one is wrong → $0.8*0.5=0.4$

200 --------→ first one is wrong and second one is correct → $0.2*0.5=0.1$

300 -------→ both are correct --→ $0.8*0.5=0.4$

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This question depends on which question we are attempting at first ->

Say it start with question 1 then question 2:-

question 1 is wrong -> amout =0 -> probality=$0.2$

question 1 is correct and question 2 is wrong -> amount =100 ->probability =$0.8*0.5=0.4$

both question 1 and 2 are correct -> amount =$300$ -> probablity =$0.8*0.5=0.4$

So expected amont if he start with question 1-> $0*0.2+100*0.4+300*0.4=160$

Say it start with question 2 then question 1:-

question 2 is wrong -> amout =0 -> probality=$0.5$

question 2 is correct and question 1 is wrong -> amount =200 ->probability =$0.2*0.5=0.1$

both question 1 and 2 are correct -> amount =$300$ -> probablity =$0.8*0.5=0.4$

So expected amont if he start with question 1-> $0*0.5+200*0.1+300*0.4=140$

So he has better chance of gettiing more money if he choose question 1 first then question 2 which will fetch him 160 ruppes .

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Now if we assume selection of question is random then ,

Selecting question no 1 first fetch us expected 160 ruppes with probability 0.5.

Selecting question no 2 first will fetch us expected 140 ruppes with probabilily 0.5 .

So ,Expected value we get = 0.5*160+0.5*140=150 ruppes .