(Problem)
A fishing boat has 10 worms and 10 leaches as bait. The bait is chosen at random. Find the probability that the 5th worm is drawn as the 6th draw from the container. Assume that there is replacement.
I'm wondering how to set these types of problems up. The answer isn't important as I have a few pages of homework that have similar questions.
I thought about setting it up as $\frac{(10 C 1)^5 }{(20 C 5)}$ but I'm unsure if that is the correct process.
WithOUT replacement.
For the 6th bait to be the 5th worm drawn, then 4 worms and 1 leech must be drawn first, and 5 worms and 9 leeches drawn after, in some order.
So you need to count the ways to select 5 worms and 1 leech, the ways to arrange 4 worms and 1 leech, and the ways to arrange the remaining 5 worms and 9 leech. Divide the product of that by the total of all ways to arrange all the bait.
===
WITH replacement.
For the 6th bait to be the 5th worm drawn, then 4 worms and 1 leech must be drawn first. The probability of drawing a worm is a constant $1/2$; assuming each bait is replaced with the same type when drawn. Here you don't need to worry about what happens afterward.
So you just need to count the ways to arrange 5 worms and 1 leech, times the 5th power of trial probability of drawing a worm, times the trial probability of drawing a leech.