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IF there be four playing at Whist, it is 15 to 1 that any two of them shall not have the four Ho∣nours, which I demonstrate thus:
Suppose the four Gamesters be A, B, C, D: If A and B had, while the Cards are a dealing, already got three Honours, and wanted only one, since it is as probable that C and D will have the next Honour, as A and B; if A and B had laid a Wager to have it, there is due to them but ½ of the Stake: If A and B
wanted two of the four, and had wager'd to have both those two, then they have an equal Hazard to get nothing; if they miss the first of those two, or to put themselves in the former Case if they get it; so they have an equal Hazard to get nothing or ½, which, by Prop. 1. is worth ¼ of the Stake; so if they want three Honours, you will find due to them 1/8 of the Stake; and if they wanted four, 1/16 of the Stake, leaving to C and D 15/16; so C and D can wa∣ger 15 to 1, that A and B shall not have all the four Ho∣nours.
It is 11 to 5 that A and B shall not have three of the four
Honours, which 1 prove thus:
It is an even Wager, if there were but three Honours, that A and B shall have two of these three, since 'tis as probable that they will have two of the three, as that C and D shall have them; consequently, if A and B had laid a Wager to have two of three, there is due to them ½ of the Stake. Now suppose A and B had wager'd to have three of four, they have an equal Ha∣zard to get the first of the four, or miss it; if they get it, then they want two of the three, and consequently there is due to them ½ of the Stake; if they miss it, then they want three of the three, and consequently
there is due to them 1/8 of the Stake; therefore, by Prop. 1. their Hazard is worth 5/16, lea∣ving to C and D 11/16.
A and B playing at Whist a∣gainst C and D; A and B have eight of ten, and C and D nine, and therefore can't reckon Honors; to find the proportion of their Ha∣zards.
There is 5/16 due to C and D upon their hazard of having three of four Honours; but since A and B want but one Game, and C and D two, there is due to C and D but ¼, or 4/16 more upon that account, by Prop. 4. this in all makes 9/16,
leaving to A and B 7/16; so the hazard of A and B to that of C and D, is as 9 to 7.
In the former Calculations I have abstracted from the small difference of having the Deal and being Seniors.
All the former Cases can be calculated by the Theorems laid down by Monsieur Hugens; but Cases more compos'd re∣quire other Principles, for the easie and ready Computation of which, I shall add one Theorem more, demonstrated after Mons. Hugens's Method.