Question

Two 4-sided dice are tossed and the sum of the down sides noted. A 4-sided die doesn’t have an “up” face like a 6-sided die has, so the down side is used. Determine the expected value of the sum. Provide evidence of your work by typing out your full solution. 

Answer 1

Below is probability distribution for sum of  down sides :

$P(X=2) =P(both \space sides \space show \space 1) =\frac{1}{16} \\ (one \space outcome \space (1,1) out \space of 4*4 =16 \space outcomes)\\ P(X=3) =\frac{2}{16} =\frac{1}{8} (two \space outcomes (1,2),(2,1))\\ P(X=4) =\frac{3}{16} (three \space outcomes (1.3),(2,2),(3,1))\\ P(X=5)=\frac{4}{16} =\frac{1}{4} (four \space outcomes (1.4),(2,3),(3.2),(4,1))\\ P(X=6) =\frac{3}{16} (three \space outcomes (2,4),(3,3),(4,2))\\ P(X=7)=\frac{2}{16} =\frac{1}{8} (two \space outcomes (3,4),(4,3))\\ P(X=8)=\frac{1}{16} (one \space such \space outcome (4,4))\\$

The expected value

$E(s)= \sum_{i=1}^{N}s_ip(s_i)\\ =2(\frac{1}{16})+3(\frac{2}{16})+4(\frac{3}{16})+5(\frac{4}{16})+6(\frac{3}{16})+7(\frac{2}{16})+8(\frac{1}{16})\\ = (\frac{80}{16})\\ =5$