Can someone please explain this to be, so that I can understand this problem?
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.
Let's say we have drawn five-card hands A, B and C (without replacement). This means that there are no duplication of cards in each hand.
The number of ways of choosing the exact 15 cards from a 52-card deck is D1=C(52,15), where
C(n,r)=n!/(r!(n-r)!)
The number of ways the 15 cards can be arranged into the three given decks A,B,C is
D2=C(15,5)*C(10,5)*C(5,5)/3!
We divide by 3! because the order of hands does not count, and there are 6 ways the hands can be formed (ABC,ACB,BAC,BCA,CAB,CBA).
Thus the probability of drawing the given 3 five-card hands is
1/(D1*D2)
=1/(2598960*126126)
=3.05×10-12
To understand this problem, let's break it down step by step.
Step 1: Selecting the three different five-card combinations from the deck.
In your favorite card game, you can select any three different combinations of five cards from the standard 52-card deck. You can do this by choosing five cards from the deck without replacement, meaning once a card is selected, it is not placed back into the deck before selecting the next card.
Step 2: Understanding dependent probabilities.
Dependent probabilities refer to the probability of an event occurring based on the outcome of a previous event. In this case, we are calculating the probability of drawing specific hands (card combinations) from the deck, which are dependent on the previous cards drawn.
Step 3: Determining the odds of drawing the specific hands directly from the deck.
To find the odds (probability) of drawing a specific hand directly from the deck, you need to calculate the number of favorable outcomes (desired hands) divided by the total number of possible outcomes (all possible combinations of five cards).
For example, let's say you want to find the probability of drawing a hand consisting of five hearts. The number of favorable outcomes would be the number of ways to choose five hearts from the deck, which is "C(13,5)" since there are 13 hearts in the deck. The total number of possible outcomes is the total number of ways to choose any five cards from the 52-card deck, which is "C(52,5)".
By dividing the number of favorable outcomes by the total number of possible outcomes, you can determine the odds or probability of drawing that specific hand.
Repeat this process for the other two hands you selected.
Step 4: Determining the probability of not drawing the specific hands directly from the deck.
To find the probability of not drawing a specific hand, you subtract the probability of drawing that hand from 1 (since the sum of all possible outcomes is 1).
For example, if the probability of drawing a specific hand is 0.2 (or 20%), then the probability of not drawing that hand would be 1 - 0.2 = 0.8 (or 80%).
Repeat this process for the other two hands you selected.
By following these steps, you can calculate both the odds of drawing specific hands and the probabilities of not drawing them directly from the deck in your favorite card game.