## Have a Heart!

It is always interesting when a probability question produces a counter-intuitive result, and the following "glimpse a heart" question is a wonderful example of that:-

Andy and Barb are a couple at a casino, gambling on getting two aces in a pair dealt face down from a (well-shuffled) standard deck of 52. At one dealing Andy happened to glimpse that one of the cards was a heart, and whispered to Barb, "Hey, let's up our bet, because the probability of two aces given at least one heart in the pair is a bit higher now (the house is playing this one as a fair game, and Andy thinks if they bet big when they get a glimpse like this then they can beat the house in the long run). Barb replied, "Don't be stupid Andy, our chances are exactly the same, I mean what if you saw a club or some other suit, you only saw the suit, it's not as if you saw one of the cards was an ace!"

They missed their chance to up the bet and didn't accidentally see any more hearts, but later on after they went home Andy (who is a bit of a math geek) showed Barb (who is supporting him through school) how the same problem had been solved by students in a discussion near the end of the ‘Playground’ problems section of the latest ‘Math Horizons’ magazine (see the section headed “Five more minutes, kids!” for the argument he showed her). The calculations (which Barb didn't really understand) showed that the conditional probability for getting two aces given at least one heart is actually 1/195 while the unconstrained probability is only 1/221

So who do you think is right in this?