Question

An urn contains 5 balls, two of which are marked $1, two $4, and one $10. A player is to draw two balls randomly and without replacement from the urn. Let X be the sum of the amounts marked on the two balls. Find the expected value of X.

Side note:

I know Rx={2,5,8,11,14}

and P(X=2) = 1/10 ... etc

I just need a proper explanation for finding the values of P(X=2), P(X=5), etc. Thanks!

Answer 1

given,

the urn contains five balls (two-S1,two S4 and one S10)

A player draws two balls randomly without replacement.

now a player can draw two balls from the urn in

ways.

therefore the number of possible draw = 10

now let us assume that S1=1, S4=4, S10=10

X= the sum of two numbers drawn

we can draw[ S1,S1](1,1) in one way out of the 10 possible ways[(S1,S1)](since there are 2 S1 ).

so P(X=S1+S1)=P(X=2)=1/10

we can draw S1,S4(1,4) in 4 ways out of the 10 possible ways.[(S1,S4),(S1,S4),(S4,S1),(S4,S1)](since there are 2 S1 and 2 S4).

so P(X=S1+S4)=P(X=5)=4/10=2/5

we can draw S1,S10(1,10) in 2 ways out of the 10 possible ways.[(S1,S10),(S1,S10)](since there are 2 S1 and 1 S10).

so P(X=S1+S10)=P(X=11)=2/10=1/5

we can draw S4,S4(4,4) in one way out of the 10 possible ways.[(S4,S4)](since there are 2 S4)

so P(X=S4+S4)=P(X=8)=1/10

we can draw S4,S10(4,10) in 2 ways out of the 10 possible ways.[(S4,S10),(S4,S10)](since there are 2 S4 and 1 S10)

so P(X=S4+S10)=P(X=14)=2/10=1/5

so X is,