I'm learning about expected probability and have got stuck.
In summing the outcomes of rolling 2 six-sided dice, how is the probability for each event given by $p_i = \frac{1}{36} (6 – abs(7 – i))p$ please?
Obviously I can check by counting instances of each sum in a multiplication grid, but I would like some intuition for how the formula works.
Best Answer
The best way is indeed to look at the table of the outcomes. Each outcome has a probability of $\frac1{36}$.
$$\begin{array}{|c|c|c|c|c|c|}\hline \text{die 1 / die 2 } & 1 &2 &3 &4 &5 &6 \\ \hline\hline \hline 1 & 2&3 &4 &5 &6 &7 \\ \hline 2 & 3 &4 &5 &6 &7&8 \\ \hline 3 &4 &5 &6 &7&8&9 \\ \hline 4 &5 &6 &7&8&9&10 \\ \hline 5 &6 &7&8&9&10 &11 \\ \hline 6 &7&8&9&10 &11 &12 \\ \hline\end{array}$$
If I use your notation I have:
$p_2=1, p_3=2, p_4=3,p_5=4,p_6=5,p_7=6,p_8=5,p_9=4,p_{10}=3,p_{11}=2,p_{12}=1$
We see a maximum is at $p_7=6$. Right and left from $p_7$ the number of values decrease by 1. So we subtract the absolute value $6-|A|$.
The value of this has to be 6 if $i=7$. Thus $|A|=0$
The value of this has to be 5 if $i=8$. Thus $|A|=1$
The value of this has to be 5 if $i=6$. Thus $|A|=1$
This should be enough to evaluate the expression for $|A|$: $|7-i|$
So in total the probability that the sum of two dice is $i$ is $\frac1{36}\cdot (6-|7-i|)$