There are 7 empty boxes numbered sequentially from 1 to 7. A sack of gold is placed into one of the boxes at random, and, independently at random, a snake is put into one of the seven boxes. Let $E_1$ be the event that the snake and the sack of gold are in different boxes, and let $E_2$ be the event that the sack of gold is not in boxes 1,2,3. What is the conditional probability that the snake is in one of the three boxes 3,4 or 5, given the event $E_1\cap E_2$?
Warning: the symmetry principle should be applied with caution in this problem, it may be safer to completely avoid using this principle here
the GOLD is in either box 4, 5, 6, 7
and the SNAKE is NOT in the same box as the GOLD
what is the probability the SNAKE is is either box 3, 4, 5
so I'm thinking that (4 choose 1) for choosing which box the gold is in...
and then there is 6 more boxes to choose for the snake because it'snot in the box with the gold so (6 choose 1)
these two facts combined might make up the denominator?
and for the numerator, it would be (4 choose 1) for where gold goes, and then (3 choose 1) for where the snake goes (either 3, 4, 5)...
but this does not seem right because snake and gold could be in the same box using my technique for the numerator