(Probability) If Nick has chosen burrow number one, what is the probability that the Easter bunny showed him burrow number three?

The fact he choose #1 is given. In your calculations, there is no need to include the ¼.

There is a prior probability (before any more information is revealed) of 1/4 that the egg is in any burrow.

If the egg is in #1 (probability 1/4) then by the rules of the game the Easter Bunny can show Nick burrow #2, #3 or #4. So there is then a 1/3 probability of showing burrow #3. Hence the 1st term of (1/4)x(1/3).

If the egg is in burrow #2, the Easter Bunny can show Nick burrow #3 or #4. There is a 1/2 chance of showing #3 in this case. Hence 2nd term is (1/4)x(1/2)

If the egg is in burrow #3, showing #3 is not an option (probability = 0). So the probability in this case is (1/4)x(0).

If the egg is in burrow #4, the Easter Bunney can show burrow #2 or #3. There is again a 1/2 probability, so in this case the overall probability is (1/4)x(1/2).

Therefore the overall probability is
(1/4)(1/3)+(1/4)(1/2)+(1/4)(0)+(1/4)(1/2)
= (1/4)(1/3+1/2+1/2+0)
= (1/4)(4/3)
= 1/3.

There isn't a formula for all possible situations. This is not a straightforward one where you just stick some numbers into a formula. To begin with, we do not know where the egg is, and because the game has already commenced, the options are limited. And the major defining fact is which burrow the egg is really in (not which guess Nick has made). So each possibility has to be examined separately.

In the following, E1 means that the egg is in number 1, and B2 means that the bunny shows number 2, and so on. Note that the probability of the egg being in any particular burrow is ¼.

So,

if E1, then the bunny can show any of the other 3, so P(B3) = ⅓. Total probability of this event is therefore ¼ x ⅓ = 1/12

if E2, then the bunny can show 3 or 4, so P(B3) = ½, for a total probability of ¼ x ½ = 1/8

if E3, then obviously P(B3) = 0

and if E4, then this is the same as E2, so probability of this event = ¼ x ½ = 1/8


Now just add those up.