Probability of Drawing Different 5 Card Hands from a Deck of 53 Cards (standard deck with 1 joker)

Question

I'm trying to calculate the probabilities for drawing different hands from a deck of 53 cards (standard deck with 1 joker)

The hands I am trying to calculate are:

A single pair

Three of a Kind

Two Pairs

Full house

Four of a Kind

Three of a Kind with 1 joker

Nothing(i.e. the probability of drawing nothing of value to the game)

Any hand containing one Joker (i assume minus the probability of drawing a Three of a Kind with 1 joker)

This is what I have so far but I know a few things are off

Answer 1

http://www.curiouser.co.uk/paradoxes/wildcard.htm

above should give answers you want.