In the game of blackjack played with one deck, a player is initially dealt 2 different cards from the 52 different cards in the deck. A winning "blackjack" hand is won by getting 1 of the 4 aces and 1 of 16 other cards worth 10 points. The two cards can be in any order. Find the probability of being dealt a blackjack hand. What approximate percentage of hands are winning blackjack hands? The probability is (Type an integer or a simplified fraction.) The percentage of hands that are winning blackjack hands is $\square \%$. (Round to two decimal places as needed.)
Solution
Step 1 ：In the game of blackjack played with one deck, a player is initially dealt 2 different cards from the 52 different cards in the deck. A winning 'blackjack' hand is won by getting 1 of the 4 aces and 1 of 16 other cards worth 10 points. The two cards can be in any order. We are asked to find the probability of being dealt a blackjack hand and what approximate percentage of hands are winning blackjack hands.
Step 2 ：The total number of possible hands is the number of ways to choose 2 cards from 52, which is a combination of 52 choose 2. This can be calculated as \(\binom{52}{2} = 1326\).
Step 3 ：A winning hand is one that contains 1 of the 4 aces and 1 of the 16 other cards worth 10 points. The number of winning hands is the product of the number of ways to choose 1 ace from 4 and the number of ways to choose 1 card worth 10 points from 16. This can be calculated as \(4 \times 16 = 64\).
Step 4 ：The probability of being dealt a blackjack hand is the ratio of the number of winning hands to the total number of possible hands. This can be calculated as \(\frac{64}{1326} = 0.048265460030165915\).
Step 5 ：The percentage of hands that are winning blackjack hands can be found by multiplying the probability by 100. This can be calculated as \(0.048265460030165915 \times 100 = 4.826546003016592\).
Step 6 ：Final Answer: The probability of being dealt a blackjack hand is \(\boxed{\frac{64}{1326}}\) and the percentage of hands that are winning blackjack hands is approximately \(\boxed{4.83\%}\).
