TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Bura)/Übungen 2020W/HW02.2

A pair of four-sided dice

Anna has a peculiar pair of four-sided dice. When she rolls the dice, the probability of any particular outcome is proportional to the sum of the results of each die. All outcomes that result in a particular sum are equally likely.

1. What is the probability of the sum being even?

2. What is the probability of Anna rolling a 2 and a 3, in any order ?

Lösungsvorschlag von Friday

--Friday Sa 30 Jan 2021 14:26:26 CET

To get a better understanding of this problem we should visualise the given situation with a table:

Die 1	Die 2	Sum	Probability
1	1	2	2/80
1	2	3	3/80
1	3	4	4/80
1	4	5	5/80
2	1	3	3/80
2	2	4	4/80
2	3	5	5/80
2	4	6	6/80
3	1	4	4/80
3	2	5	5/80
3	3	6	6/80
3	4	7	7/80
4	1	5	4/80
4	2	6	6/80
4	3	7	7/80
4	4	8	8/80

Part a)

Now all we have to do is to sum up the probabilities of the outcomes that result in an even sum.

$${\frac {2}{80}}+{\frac {4}{80}}+{\frac {4}{80}}+{\frac {6}{80}}+{\frac {4}{80}}+{\frac {6}{80}}+{\frac {6}{80}}+{\frac {8}{80}}={\frac {40}{80}}=0.5\quad \Box$$

Part b)

All we have to do here, is to add the probabilities of rolling (2-3) and (3-2).

$${\frac {5}{80}}+{\frac {5}{80}}={\frac {1}{8}}=0.125\quad \Box$$