poker probability a pack of poker contains 52 cards and we are going to flip through one by one so the probability of the following events are


*

*a king right after the first ace ?

*an ace right after the first ace ?

*the first ace is the 10-th card?

*the probability the next card is the ace of spades, if the  first ace is the 30-th card?

*The probability the next card is the jack of diamonds If the 
first ace is the 30-th card?


I have tried the (1)(2)(3) and not sure is right or not,but no idea with the (4) (5)
(1)$\frac{4}{52}*\frac{4}{51}$ and which is same prob as (2)
(3)$\frac{48}{52}*\frac{47}{51}*\frac{46}{50}*\frac{45}{49}*\frac{44}{48}*\frac{43}{47}*\frac{42}{46}*\frac{41}{45}*\frac{40}{44}*\frac{4}{43}$but any simpler way to write it out?

1) King follows First Ace.
Let's not worry about suits or values of the other cards.  Take a pack of four king of hearts, four ace of spades, and forty four jokers.
There are $52!/(4!4!44!)$ equally probable ways to arrange this deck.
Set aside one king, shuffle the remaining cards, stick that king after the first ace.  There are $51!/(3!4!44!)$ ways to do this.
So the probability of the event is $\quad\dfrac 1 {13}$

2) Ace follows First Ace.
As above, so below.  
Set aside one ace.   There are $51!/(4!3!44!)$ ways to sort 4 kings, 3 aces, 44 jokers (and stick the set aside ace behind the first ace).   Coincidentally yielding the same probability of $1/13$.

3) First Ace is tenth card.
Take 4 ace of spades and 48 jokers.  $52!/(4!48!)$ ways to arrange them in total.  Now count ways to arrange 9 jokers, an ace in tenth place, and then all the rest (3 aces, 39 jokers).

4) Ace of spades follows first ace when that is in 30th place.
The deck consists of three ace of hearts, one ace of spades, and 48 jokers.  The deck is shuffled so we have 29 jokers, an ace, in that order, and then the rest (3 aces, 19 jokers) in any order.
If the first ace is spades, it cannot follow itself.   Multiply the probability that it is not, by the conditional probability of the event given that.

5) Jack of Diamonds follows First Ace when that is in thirtieth place.
The deck consists of four ace of hearts, jack of diamond, and 47 jokers, shuffled so we have 29 cards, an aces, and then the rest.  
If the jack is among the 29 cards before first ace, then it cannot follow it.   Multiply the probability that it is not, by the conditional probability of the event given that.
