Select three different five card combinations or five-card hands from your favorite card game that utilizes a standard 52-card deck containing four suits (clubs, hearts, diamonds, and spades), with each suit containing 13 cards with numbers 2-10 and face cards ace, king, queen, and jack. Then, do the following.
Using the concept of dependent probabilities, determine the odds that you would draw these hands (card combinations) directly from a deck of cards.
Determine the probability that you would not draw these hands (card combinations) directly from a deck of cards.
This is what I got for the first part and I feel it's wrong. Didn't work in the second part yet.
When looking at the possibilities of drawing 5 cards from a deck of 52 there are going to be 2,598,960 possibilities. Now if we are looking to pull 2 kings, a heart, and 2 10’s, it would be 4/52, 3/51, 4/50, 4/49, 3/48 = (4*3*4*4*3/52*51*50*49*48) = The probabilities these cards would be pulled in a five-card combination is 576/ 311,875,200 =1/541,450 or 1.86E-06
Here is the answer I got, am I on the right track?
Have you considered the fact the the king and/or the 10 could be a heart? Consider those probabilities.
Either-or probabilities are found by adding the individual probabilities.
To determine the probability of drawing specific hands from a deck of cards, you need to calculate the number of favorable outcomes (combinations that meet the criteria) and divide it by the total number of possible outcomes. Let's begin by finding three different five-card combinations or hands from a standard deck of 52 cards.
1. Royal Flush: This is the highest-ranking hand in many card games and consists of a Ten, Jack, Queen, King, and Ace of the same suit. There are four suits in a deck, so there are four possible Royal Flush combinations.
2. Four-of-a-Kind: This hand comprises four cards of the same rank and one "kicker" card. We can consider the Four-of-a-Kind with Aces as an example. There are four Aces in a deck, so we choose four of them for our hand. For the fifth card, we can pick any of the remaining 48 cards in the deck. Thus, the number of Four-of-a-Kind combinations is (4 choose 4) * (48 choose 1), where "choose" represents the combination formula. You can calculate this value to get the exact number.
3. Full House: A Full House consists of three cards of the same rank and a pair of cards with another rank. Let's use three Queens and two Fives as an example. There are four Queens in a deck, so we choose three of them for our hand. For the two Fives, there are four Fives in the deck, and we choose two of them. The number of Full House combinations is (4 choose 3) * (4 choose 2).
Now let's address the probability calculations:
1. To find the probability of drawing a specific hand like the Royal Flush, divide the number of favorable outcomes (4 Royal Flush combinations) by the total number of outcomes for a five-card hand from a deck of 52 cards. The total number of outcomes is given by (52 choose 5).
Probability of drawing a Royal Flush = (Number of Royal Flush combinations) / (Number of total combinations) = 4 / (52 choose 5).
2. The probability of not drawing a Royal Flush would be equal to 1 minus the probability of drawing a Royal Flush. So, the probability of not drawing a Royal Flush = 1 - [4 / (52 choose 5)].
You can apply a similar reasoning to calculate the probabilities for the other two types of hands (Four-of-a-Kind and Full House) by substituting the respective number of favorable outcomes for each hand.
Remember, the formula for calculating probabilities is the number of favorable outcomes divided by the total number of possible outcomes.