What is the expected value of the game if 3 hits pay​ $6, 2 hits pay​s $4, 1 hit pays $2, and 0 hits pay​s costs $4?

The probability of getting 3 hits is
     [tex]P\left(3\:hits\right)=0.7\times 0.5\times 0.3=0.105[/tex]

The probability of getting 2 hits is 
     [tex]P\left(2\:hits\right)=\left(0.7\times 0.5\times 0.7\right)+\left(0.7\times 0.5\times 0.3\right)+\left(0.3\times 0.5\times 0.3\right)=0.395[/tex]

The probability of getting 1 hit is 
     [tex]P\left(1\:hit\right)=\left(0.7\times 0.5\times 0.7\right)+\left(0.3\times 0.5\times 0.7\right)+\left(0.3\times 0.5\times 0.3\right)=0.395[/tex]

The probability of getting 0 hit is 
     [tex]P\left(0\:hit\right)=0.3\times 0.5\times 0.7=0.105[/tex]

The expected value is solved by adding the products of the probability by the given pay. That is 
   [tex]E=0.105\left(6\right)+0.395\left(4\right)+0.395\left(2\right)+0.105\left(4\right)=3.42[/tex]
     
The expected value is $3.42.