Let's tackle this problem step by step and explain each part clearly.
### a. Finding the number of ways to draw any two cards from a 52-card deck
In this problem, we need to determine how many ways we can choose 2 cards from a deck of 52 cards. This is a combination problem, often denoted as [tex]\(\binom{n}{k}\)[/tex], where [tex]\(n\)[/tex] is the total number of items, and [tex]\(k\)[/tex] is the number of items to choose.
For our case:
- [tex]\(n = 52\)[/tex] (total cards in a deck)
- [tex]\(k = 2\)[/tex] (cards to draw)
The number of ways to choose 2 cards from 52 cards is given by:
[tex]\[
\binom{52}{2} = \frac{52!}{2!(52-2)!} = 1326
\][/tex]
So, there are 1326 ways to draw any two cards from a deck of 52 cards.
### b. Finding the number of ways to draw two valuable blackjack cards
Next, we need to find out how many ways we can draw 2 cards from the 20 valuable cards (Jacks, Queens, Kings, Aces, or Tens). Similar to the first part, this is also a combination problem.
For our case:
- [tex]\(n = 20\)[/tex] (total valuable cards)
- [tex]\(k = 2\)[/tex] (cards to draw)
The number of ways to choose 2 valuable cards from the 20 valuable cards is given by:
[tex]\[
\binom{20}{2} = \frac{20!}{2!(20-2)!} = 190
\][/tex]
So, there are 190 ways to draw two valuable blackjack cards from the 20 valuable cards.
### c. Calculating the probability that two valuable cards are dealt
To find the probability that two valuable cards are dealt, you take the number of favorable outcomes (drawing two valuable cards) and divide it by the number of all possible outcomes (drawing any two cards from the deck):
[tex]\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{190}{1326}
\][/tex]
The calculated probability is:
[tex]\[
\frac{190}{1326} \approx 0.1433
\][/tex]
Therefore, the probability that two valuable cards are dealt during a game of blackjack is approximately [tex]\(0.1433\)[/tex] or [tex]\(14.33\%\)[/tex].
In summary:
- The number of ways to draw any two cards from a 52-card deck is [tex]\(1326\)[/tex].
- The number of ways to draw two of the 20 valuable blackjack cards is [tex]\(190\)[/tex].
- The probability that two valuable cards are dealt is approximately [tex]\(0.1433\)[/tex] or [tex]\(14.33\%\)[/tex].
