## Playing the Odds: How Many Boys and Girls?

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You've decided to breed a particular fish and it's expensive, so you plan to order very young fish and hope you get the right number of males and females. What are your odds? In the tables below I have calculated the odds of various desirable combinations of males and females, depending upon the size of your order. In this I have made two assumptions, either or both of which may be false in any given case.

1. The young hatch at 50:50 male:female ratio.
2. The dealer or breeder ships a random selection to you.

I once bought a entire LFS's lot of 12 scrawny, bedraggled little Apistogrammus sp. (dwarf cichlids) and every one of the 12 turned out to be male. I find it hard to believe that all the eggs that hatched in that batch were males, so I have to assume that the dealer involved was shipping only males to the LFS. The reason could have been benign (males have more color and might sell better) or malign (obviously without females the hobbyists can't become competition for the breeders). Whatever the reason I took all 12 of the now well-grown and very belligerent male Apistogramma sp. back to the LFS and exchanged them for a known pair of pearl gouramis.

Summary Table

 # Fish Bought 2 Quartets 1 Quartet and 1 Trio 2 Trios Quartet Trio Group with 1 Male Unbreedable Group 2 NA NA NA NA NA 50 (1 female) 50 3 NA NA NA NA 37.5 37.5 (2 females) 25 4 NA NA NA 25 62.5 25 (3 females) 12.5 5 NA NA NA 46.9 78.1 15.7 (4 females) 6.2 6 NA NA 23.4 64 87.4 9.4 (5 females) 3.1 12 61.0 80.3 92.7 98.1 99.7 0.3 (11 females) 0

If you only want a pair of fish, look to the "Unbreedable Group" column. This is your chance of failing to get even one pair in the number of fish bought.

If you are breeding a type of fish that requires more than one female per male, then the other columns apply to you.

I have seen it recommended that two people buy a larger shipment and split the fish, as means of both acquiring what they want with a higher percent probability. If you want to split a shipment with someone else, then you have to devise your deal carefully or your chances of getting what you want may be at best no better than if you bought a smaller shipment by yourself, and you may have more unwanted fish to dispose of.

For instance, suppose you both want a trio of fish. If you each buy 6 fish, then you each have an 87% chance of getting a trio and 3 fish to dispose of if you do get it. Splitting a shipment of 12 you each have a 92% chance of both getting a trio, and whoever ordered has 6 fish to dispose of. If you both want a quartet, then you each have a 64% chance of getting it with 6 fish, and you each have 2 fish to dispose of if you do get it. Splitting a shipment of 12 you each have only a 61% chance of both getting a quartet, and whoever ordered has 4 fish to dispose of.

If, however, you want a quartet and you can persuade your partner to accept a trio, then you have a much better (98 versus 64) chance of getting your quartet by ordering 12. Your partner has less of a chance of getting a trio (80 versus 87) with this order size when you are requiring a quartet for yourself.

I'm not suggesting that you should cheat someone, nor that someone is intentionally trying to cheat you, with a shipment splitting deal. What I'm pointing out is that the only way this deal can benefit anyone with regards to the ratio of males and females is if the person doing the ordering plays the role of distributor and takes first crack at the fish received before offering the rest up for sale to other parties. It is neither surprising nor entirely unfair that the greater benefit in this goes to the person who places the order and receives in the fish. The benefit of lower shipping costs remains, and in fact can be included as an inducement to the junior partner(s) in the deal to accept any less desirable breeding groups.

Raw Data Table

Percent Chance of Number of Females as a Function of Number of Fish Bought; percentages are rounded and not therefore strictly accurate. For example, it is not really true that there is 0 percent chance of getting either combination at the ends of the 12-fish entry, but the probability is below 0.1 percent.

 Females Fish Bought Combos 0 1 2 3 4 5 6 7 8 9 10 11 12 2 4 1= 25% 2= 50% 1= 25% 3 8 1= 12.5% 3= 37.5% 3= 37.5% 1= 12.5% 4 16 1= 6.25% 4= 25% 6= 37.5% 4= 25% 1= 6.25% 5 32 1= 3.1% 5= 15.7% 10= 31.2% 10= 31.2% 5= 15.7% 1= 3.1% 6 64 1= 1.6% 6= 9.4% 15= 23.4% 20= 31.2% 15= 23.4% 6= 9.4% 1= 1.6% 7 128 1= 7= 21= 35= 35= 21= 7= 1= 8 256 1= 8= 28= 56= 70= 56= 28= 8= 1= 9 512 1= 9= 36= 84= 126= 126= 84= 36= 9= 1= 10 1024 1= 10= 45= 120= 210= 252= 210= 120= 45= 10= 1= 11 2048 1= 11= 55= 165= 330= 462= 462= 330= 165= 55= 11= 1= 12 4096 1= 0.0% 12= 0.3% 66= 1.6% 220= 5.4% 495= 12.1% 792= 19.3% 924= 22.6% 792= 19.3% 495= 12.1% 220= 5.4% 66= 1.6% 12= 0.3% 1= 0.0%