Independent solution

How to solve this Independence question

Setup

Setup

Determine the distribution of the total on a single trial from the two face-value probabilities.

Pr⁡(D=1)=13,Pr⁡(D=2)=23\Pr(D=1)=\frac13,\qquad \Pr(D=2)=\frac23
Pr⁡(S=2)=19,Pr⁡(S=3)=49,Pr⁡(S=4)=49\Pr(S=2)=\frac19,\qquad \Pr(S=3)=\frac49,\qquad \Pr(S=4)=\frac49

Model

Model

It is shorter to calculate the probability that the two independent trial totals match, then take its complement.

Pr⁡(S1=S2)=∑s=24Pr⁡(S=s)2\Pr(S_1=S_2)=\sum_{s=2}^{4}\Pr(S=s)^2

Compute

Compute

Square and add the three single-trial probabilities.

Pr⁡(S1=S2)=(19)2+(49)2+(49)2=3381\Pr(S_1=S_2)=\left(\frac19\right)^2+\left(\frac49\right)^2+\left(\frac49\right)^2=\frac{33}{81}
Pr⁡(S1≠S2)=1−3381=4881=1627\Pr(S_1\ne S_2)=1-\frac{33}{81}=\frac{48}{81}=\frac{16}{27}

Answer

Answer

The probability of different totals is sixteen twenty-sevenths.

1627(C)\boxed{\frac{16}{27}\quad\text{(C)}}