## Have a Heart!

ok – Let’s see who stands where here.

As posed by Bruce Aubertin:

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!”

Who is right in this?
(Barb is supporting Andy, a bit of a math geek, through school.)

The above was prompted by a discussion near the end of the ‘Playground’ problems section of the latest ‘Math Horizons’ magazine (see the section headed “Five more minutes, kids!”).

So, what do you think? Click Here to vote (password is “glimpseheart”)

### One Comment

1. Alan Cooper says:

Later Andy expalined that he had approached the problem using conditional probability, as follows:

The glimpse tells us that there is at least one heart, so if A is the event that we had two aces and H is the event that we had at least one heart, then what we need to know is whether P(A) and P(A|H) differ.

The latter quantity is a conditional probability, the probability that A occurs given that H occurs; it can be computed using the formula P(A|H) = P(A&H) / P(H), where A&H is the event that the opponent has both two aces and at least one heart.

After computing the probabilities P(A&H)=1/442 and P(H)=15/34, it follows that P(A)=1/221 and P(A|H)=1/195, which is more than 1/221.

So if there is at least one heart then the probability of getting two aces is higher.

Barb said she didn’t understand this “conditional probablity” stuff and demanded a simpler explanation because her “intuition” said that there should be no difference.