Given a standard deck of 52 cards, what is the probability of drawing four cards in a row that form a two-pair poker hand?

Official Answer

Certainly, approaching a question like the probability of drawing four cards that form a two-pair hand from a standard deck of 52 cards requires a blend of analytical thinking and a deep understanding of probability theory, both of which are integral to my role as a Data Scientist.

To begin, let's break down the problem into more manageable parts. A two-pair hand in poker means having two cards of one rank, two cards of another rank, and one card of a third rank. However, since we're focusing on drawing just four cards, we're looking at getting exactly two pairs of different ranks.

The first step is to calculate the total number of ways to draw four cards from a deck of 52. This is a classic combination problem, where the order does not matter, calculated as "52 choose 4". The formula for combinations is C(n, k) = n! / (k!(n-k)!), where n is the total number of items, k is the number of items to choose, and "!" denotes factorial. So, for our case, C(52, 4) = 52! / (4!(52-4)!) = 270,725 possible combinations of four cards.

Next, we need to calculate the number of combinations that result in a two-pair hand. To do this, we first select the ranks of the pairs. There are 13 ranks in a deck, and we need to choose 2 different ranks for our pairs, which can be done in "13 choose 2" ways. Then, for each of these pairs, there are "4 choose 2" ways to choose the two cards from the four available suits. Since we have two pairs, we square this value.

Putting it all together, the number of ways to get a two-pair hand is C(13, 2) * (C(4, 2))^2. C(13, 2) calculates to 78, and C(4, 2) calculates to 6. Squaring C(4, 2) gives us 36, and multiplying this by 78 gives us 2,808 unique ways to draw a two-pair hand from a deck of 52 cards.

Therefore, the probability of drawing a two-pair hand in four cards is the number of favorable outcomes divided by the total number of outcomes, which is 2,808 / 270,725.

Simplifying this fraction gives us approximately 0.0104, or 1.04%. This means that if you draw four cards from a standard deck, there's about a 1.04% chance you'll end up with a two-pair hand.

In my experience as a Data Scientist, approaching problems methodically, breaking them down into smaller, more manageable parts, and applying fundamental statistical principles, has been key to not only solving complex problems but also to explaining these solutions in an accessible way. This approach allows me to contribute effectively across diverse teams and projects, and it's how I would tackle challenges and opportunities in this role.