For my year 11 specialist math PSMT, I've been tasked with designing a card game with 5 divisions. The game I designed was "5 Card Draw" (working title), it is played with a deck of 53 cards (standard deck with 1 additional joker) where the player simply draws a hand of 5 cards. The outcomes of the game are as follows: The player can either.
Draw Nothing or a Joker (House Wins): By drawing nothing I mean drawing something that is not a hand of significance in the context of my game. By drawing a joker the rest of the hand does not matter but this excludes the probability of drawing Three of a Kind w/ a Joker as it is an exception.
A Single Pair (Division 5): The probability of drawing two cards of the same number or face, ignoring the possibility of a drawing a Joker with the pair in the probability as this nullifies the significance of the pair in the context of my game. (this is the same case for the rest of the hands)
Two Pairs (Divison 4): The probability of drawing two pairs, ignoring the possibility of a drawing a Joker with this...
Three of a Kind (Divison 3): The possibility of drawing a Three of a Kind, ignoring the possibility of a drawing a Joker with this...
A Full House (Divison 2): The possibility of drawing a Full House
A Four of a Kind or A Three of a Kind w/ a Joker (Divison 1): The Probability of Drawing a four of a kind, ignoring the possibility of a drawing a Joker with this... As well as drawing Three of a Kind w/ a joker.
My current set of calculations is here: 
The sum of my calculations is roughly 96%
My problem is a few of the calculations must be off as their sum doesn't add to give 100% and this would usually be ignored if the gap wasn't as substantial meaning one or more of the calculations are wrong
I'm hoping I can be assisted with these calculations and any help would be greatly appreciated.
In the last division, you only counted the hands with the joker and $4$ different ranks. None of the counts seems to include the hands with the joker and a pair, two pairs or four of a kind.
Everything else looks right to me.
Edit in response to the comment:
I’m surprised you couldn’t figure it out because you were doing very well with all the other cases, and it’s basically the same problem that you solved for the case of the joker and three of a kind:
joker and one pair: $\frac{\binom{13}1\binom42\binom{12}2\binom41^2\binom11}{\binom{53}5}=\frac{6336}{220745}\approx2.87\%$,
joker and two pairs: $\frac{\binom{13}2\binom42^2\binom11}{\binom{53}5}=\frac{216}{220745}\approx0.10\%$,
joker and four of a kind: $\frac{\binom{13}1\binom44\binom11}{\binom{53}5}=\frac1{220745}\approx4.53\cdot10^{-4}\%$.
Since your probabilities in fact add up to $97\%$, not $96\%$, the total comes out right.