Probability of dice rolls

Question

Two regular cubic dice are rolled. One is red and the other blue. What s the probability that the red die roll is greater that or equal to the blue die?

Answer 1

We only have to consider the case there they are equal, as otherwise one will be greater than the other.

There are 36 outcomes in total (6$\times$6). There are 6 outcomes that will be the same (1,1), ... , (6,6) $\rightarrow$ $\frac{1}{6}$. Therefore, the chance of them not being equal is $\frac{5}{6}$. Half of those will be in favor of the red die, i.e. $\frac{5}{6} : 2$ = $\frac{5}{12}$.

To get greater or equal, just add the $\frac{1}{6}$, which gives us $\frac{7}{12}$.

Answer 2

Hints:

• $P(R\geq B)=P(B\geq R)$ because of symmetry.
• $P(R\geq B)+P(B\geq R)=1+P(B=R)$ (do you understand why?)

So you are ready if you can find $P(B=R)$.

Answer 3

The other way to calculate the value is to count the possibility for each dice roll by fixing one value . Fix the Blue Dice value.

(x,1) -> 6 red values which will be greater than or equal to the blue one.

(x,2) -> 5

(x,3) -> 4

(x,4) -> 3

(x,5) -> 2

(x,6) -> 1

So total possibilities where red >= blue are 6+5+4+3+2+1 = 21 Total possibilties = 6*6 = 36 Thus the answer is : 21/36 = 7/12