Question:
Playing 5 card majors and 15-17 NT, can you estimate the % of hands which are opened with 1♣ or 1♦?

Thanks in advance,

Marty Zucker.

Reply from Ed Hoogenkamp ('South'):

Dear Marty,

I´m escaping from this one.....Peter knows his mathematics a lot better than I do. So please read his answer

Regards from Barcelona

Reply from Peter van der Linden ('North'):

Dear Marty,

Your question is short but difficult!
Especially since Ed and I are not exactly experts in the field of mathematics/statistics. (Ed therefore passed the question on to me…).
I will give it a try anyway, using the Mathematical Tables of The Official Encyclopedia of Bridge. I had to make some assumptions (M=major suit, m=minor suit).

Assumption 1: only hands holding 12 HCP min - 19 HCP max qualify for an opening bid on the one level and, importantly: all of them (this is of course a simplification since e.g. hands holding 7+ cards in a M usually will be opened 2♣ or 4M).
This way 33,4721% of all hands qualify for an opening bid of 1♣/♦/♥/♠/NT.

Assumption 2: No 1NT opening on 5M-3-3-2

Assumption 3: All 12-19 HCP points containing a five card suit in a M are opened with 1M (also a simplification considering 6/7-5 in m-M). 

1. How many hands contain a five card suit or longer in a M?
59,303% of all hands contain one five card suit of longer, meaning 29,652% of all hands contain one five card suit or longer which is a M.
5,6119% of all hands contain two five card suits or longer, meaning 4,6766% of all hands contain two five card suits or longer of which at least one is a M (only 1 of 6 cases is m-m!). 

2. How many hands are thus opened 1M?
Of all hands 29,652 + 4,6766  =  34,329% contain at least one five card suit M or longer.
All of these (assumption 3) are opened with 1M, meaning 34,329% of all 33,4721% hands that are opened, are opened with 1M = 11,491%. 

3. How many hands are opened 1NT?
Of all hands 10,0963% hold 15-17 HCP.
Of these 4,0229% qualify for 1NT (4-4-3-2, 4-3-3-3, 5m-3-3-2). 

Conclusions
Of all hands that are opened on the one level (33,4721% of all hands):
11,491% are opened with 1M
  4,023% are opened with 1NT
17,958% are opened with 1m (33,472 – 11,491 – 4,023). 
This means the ratio of 1m : 1M is a bit higher than 3 : 2. (If you do open 1NT on 5M-3-3-2 the odds will change even more - though slightly - in favour of 1m). 

I hope this will do. If any of our visitors would like to add to (or correct…) my figures, please do! (I do not expect our man from Barcelona to oblige; he prefers playing bridge above sweating on mathematics…).

Regards from Orkanger