Dave Computes Five-Card Draw Probabilities

In this post, I'll provide most probabilities as percentages rounded to the nearest percent.
(It's harder to memorize or identify patterns with more precise numbers.)

A hand of 5 cards is dealt. How strong is it likely to be?

1% chance of straight-or-better
2% chance of trips
5% chance of 2 pair
42% chance of a pair
50% chance of high card hand

Let's break those high card hands down further. Of all hands,
50% are pairs or better
19% are A high
13% are K high
8% are Q high
5% are J high
3% are T high
1% are 9 high
0.5% are 8 high
0.15% are 7 high

There's about a 42 / 13 ≈ 3.23% chance of being dealt any particular pair.

We can therefore rank all hands as follows:

top 1%:  straight-or-better
top 3%:  trips
top 8%:  2 pair
top 11%:  AA
top 14%:  KK
top 18%:  QQ
top 21%:  JJ
top 24%:  TT
top 27%:  99
top 31%:  88
top 34%:  77
top 37%:  66
top 40%:  55
top 44%:  44
top 47%:  33
top 50%:  22
top 69%:  A high
top 82%:  K high
top 90%:  Q high
top 95%:  J high
top 98%:  T high
top 99%:  9 high
top 100%:  8 high or 7 high

Now let's consider what happens when we exchange cards.

We're dealt a pair and we exchange the other 3 cards:
71% chance unimproved
16% chance of 2 pair
11% chance of trips
1% chance of full house

We're dealt 2 pairs and we exchange the other 1 card:
91.5% chance unimproved
8.5% chance of full house

We're dealt trips and we exchanged the other 2 cards:
90% chance unimproved
6% chance of full house
4% chance of quads

There's a 5% chance we'll be dealt a high card hand with a 4-flush or open-ended straight draw.

We're dealt a 4-flush and we exchange the other 1 card:
19% chance of flush
26% chance of pair
55% chance unimproved

We're dealt an open-ended straight and we exchange the other 1 card:
17% chance of straight
26% chance of pair
57% chance unimproved

A common situation occurs when I have a pair and I'm up against a single opponent, who I believe has a higher pair. We each exchange 3 cards. To win, I need to make 2-pair or better, and I need my opponent to remain unimproved. I've got a 29% change of improving, and my opponent has a 71% of not improving. Overall, that's a 29% × 71% ≈ 21%. That's very close to the probability that I make my 4-flush or open-ended-straight draw.

On the other hand, if I make a straight or flush draw, it's much more likely to hold up in a multi-player hand. If I'm one of 3 players holding a pair, then now I only have a 29% × 71% × 71% ≈ 15% chance of overtaking my opponents. That number falls to 10% if 4 players exchange.