Meet Donny Notrump (part 2)

Number Theory Level 2

Donny ("Three") Notrump, precocious 10-year old logician, mathematician-in-training and expert bridge player, was telling me about a hand he played of his favorite card game.

"And I felt pretty confident. I had the A-K-10-9 of clubs in my hand and two low clubs in the dummy. I could take two finesses, and as long as West didn't have both the queen and the jack of clubs, one of the finesses at least would work and I would make my bid. My chances of winning were..."

Finish Donny's statement.

74% 75% 76% 87.5%

Denton Young by Denton Young , 47, USA

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1 solution

Denton Young
Feb 1, 2017

It's easier to work backwards. East and West together have 26 cards.

The chance that West has the queen of clubs is 13/26. If West holds that card, there are now 25 spots available for the jack of clubs, of which 12 are in West's hand. So the chance of him holding the jack of clubs is 12/25.

Therefore, that chance of West holding both key cards is 13/26 * 12/25 = 6/25 = 0.24 = 24%.

So the chance that West does NOT hold both key cards is 1 - 0.24 = 0.76 = 76%.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

This question should probably be categorized under Discrete Math. Your logic seems to work, but what about another way of viewing it: the dealer, after looking at his hand and dummy, knows that the queen & jack are in the opponents' hands. There are 4 ways this could happen, where only 1 way has both cards in the wrong hand. All 4 cases should be equally likely. This gives an answer of 75% ???

Richard Costen - 4 years, 4 months ago

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That's why the question is trickier than it looks. The key is to remember that the queen and jack of clubs being in the same hand are not independent events.

Denton Young - 4 years, 4 months ago

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Think I have it now: West with queen & jack - 13/26 * 12/25 = 12/50 = 24% East with queen & jack - 13/26 * 12/25 = 12/50 = 24% West with queen, East with jack - 13/26 * 13/25 = 13/50 = 26% East with queen, West with jack - 13/26 * 13/25 = 13/50 = 26% Total of the 4 possibilities = 100%

Richard Costen - 4 years, 4 months ago

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@Richard Costen – Yep. That's exactly the correct way to calculate it.

Denton Young - 4 years, 4 months ago

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