What is the probability of being dealt a blackjack hand?

In the game of blackjack, a "blackjack" is typically defined as getting an Ace and a 10-valued card (10, Jack, Queen, or King) in the initial two-card hand. To calculate the probability of getting a blackjack, we must first understand we are dealing with a standard deck of 52 cards, which comprises 4 suits with 13 ranks in each suit.

There are 4 Aces in the deck and 16 cards that are worth 10 points (four tens, four jacks, four queens, and four kings). To get a blackjack, you need one Ace and one 10-point card.

Step 1: Calculate the probability of drawing an Ace first. There are 4 Aces out of 52 cards, so the probability of drawing an Ace is 4/52 or 1/13.

Step 2: After drawing an Ace, calculate the probability of drawing a 10-value card. Now there are 51 cards left, and as we figured earlier, there are 16 cards worth 10 points. The probability of drawing one of these cards is 16/51.

Step 3: Multiply the probabilities of both steps to find the overall probability for this sequence (Ace first, then a 10-value card). P(Ace then 10-value card) = (4/52) * (16/51) = (4/13) * (16/51) = 64/663

However, you can also get a blackjack by getting a 10-value card first and then an Ace.

Step 4: Calculate the probability of drawing a 10-value card first. The probability is 16/52 or 4/13, as there are 16 such cards in the deck.

Step 5: Calculate the probability of drawing an Ace after drawing a 10-value card. With 51 cards remaining and 4 Aces left, the probability is 4/51.

Step 6: Multiply the probabilities of both steps just like we did earlier. P(10-value card then Ace) = (16/52) * (4/51) = (4/13) * (4/51) = 64/663

Step 7: Add the two probabilities together because either sequence results in a blackjack. P(Blackjack) = 64/663 + 64/663 = 128/663

Step 8: Simplify the probability. 128/663 can be simplified by dividing both numerator and denominator by their greatest common divisor. However, in this case, 128 and 663 have no factors in common other than 1, so the probability is already in its simplest form.

Therefore, the probability of being dealt a blackjack from a standard deck on the first two cards is 128/663, which is approximately 19.3%. Please note that this calculation assumes the use of a single deck of cards. In games with multiple decks, the odds would differ slightly due to changes in the combinations available.