Two bullets are put into a gun's round barrel consecutively, which has capacity of 6. The gun is shot once, but no bullet is fired. Does rolling the barrel (shuffling) before next shot increase the probability of firing a bullet?
Since one empty slot has been used, there must be more probability of shooting a bullet in next chance, than shooting after a shuffle.
Since there are 6 total slots with 2 bullets, the probabilty of firing a bullet after shuffle is 2/6.
The probability of getting bullet without shuffle is 1/4. To understand this, look at the image, and imagine that one of the empty slots (3,4,5,6) just got shot. Now assumming the barrel turns clockwise, next shot will be 4 ,5 ,6 or 1. and out of these four, only (1) has bullet. Hence probability of firing a bullet is 1/4. Note that this same case stands in anticlockwise direction.
With shuffle its 2/6=1/3. Hence yes, shuffling does increase probability of shooting bullet (from 1/4 to 1/3)