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The whole roll 2n+1 times, get k even numbered die is a binomial distribution but I wonder

if the symmetry of the probability can't make this a bit easier.

We can definitely use the fact that it is multiple choice though...

If \(n=0\) then clearly getting an even face means you win, so the probability is \(1/2\).

Which knocks out b and c

i am getting d as the answer , but the book is saying a.

When \(n=1\). what is the probability?

You have three rolls, if you roll 1 or 3 evens, you win

hm,yes

oops, that doesn't actually matter, when \(n=1\) then a) is still 1/2

They need to be different.

n=2 gives 7/10 for a, but d is still 1/2 so....
5 rolls, what is the probability of odd even rolls

Weird...
There is no explanation for a)?

D i think