Probability & Statistics Ch.2 Section 3 Question 1

· Mohammad-Ali Bandzar

Solutions for “Probability and Statistics: The Science of Uncertainty” (Second Edition). These are solutions I have come up with; I offer no guarantee of accuracy.

Question

Consider rolling two fair six-sided dice. Let Y be the sum of the numbers showing. What is the probability function of Y?

Solution

When rolling 2 dice, there are 36 equally likely outcomes. Diagonals in the outcome grid (starting from any number in the leftmost column and going right one then up one) will all have the same sum. For example, (5,1) has the same sum as (4,2), (3,3), (2,4), (1,5).

pY(2)=136pY(3)=236pY(4)=336pY(5)=436pY(6)=536pY(7)=636pY(8)=536pY(9)=436pY(10)=336pY(11)=236pY(12)=136 \begin{aligned} p_Y(2) &= \frac{1}{36} \\ p_Y(3) &= \frac{2}{36} \\ p_Y(4) &= \frac{3}{36} \\ p_Y(5) &= \frac{4}{36} \\ p_Y(6) &= \frac{5}{36} \\ p_Y(7) &= \frac{6}{36} \\ p_Y(8) &= \frac{5}{36} \\ p_Y(9) &= \frac{4}{36} \\ p_Y(10) &= \frac{3}{36} \\ p_Y(11) &= \frac{2}{36} \\ p_Y(12) &= \frac{1}{36} \end{aligned}

pY(y)=0p_Y(y)=0 for all y∉{2,3,4,5,6,7,8,9,10,11,12}y\notin\{2,3,4,5,6,7,8,9,10,11,12\}