A game is played by drawing 4 card from an ordinary deck and replacing each card after it is drawn.find the probability that at least 1 king is drawn

When a problem requires "at least one", it is often easier to find the probability of getting none and subtracting the probability from 1 (complement).

Here, probability of not getting a king at a draw is 48/52, or 12/13.
Probability of getting no king at 4 consecutive draws (with replacement) is (12/13)^4.
The complement is 1-(12/13)^4, which is the probability of getting at least one king.

Well, isn't it a "royal" challenge you've got here! Let's calculate the probability of drawing at least one king from a deck.

First, let's determine the total number of possible outcomes. Since we are drawing four cards with replacement, there are 52 options for each draw. Hence, the total number of outcomes is (52^4).

Now, let's find the number of favorable outcomes, i.e., the outcomes where at least one king is drawn. There are three scenarios we need to consider: drawing one, two, or three kings.

Scenario 1: Drawing one king and three non-kings.
There are 4 ways to select which draw will have a king and (48^3) ways to select the non-king cards. So, the number of favorable outcomes for this scenario is (4 * (48^3)).

Scenario 2: Drawing two kings and two non-kings.
The number of ways to select which two draws will have kings is (4 Choose 2) = 6. Similarly, (48^2) is the number of ways to select the non-king cards. So, the total number of favorable outcomes for this scenario is (6 * (48^2)).

Scenario 3: Drawing three kings and one non-king.
There are (4 Choose 3) = 4 ways to select which three draws will have kings, and 48 ways to select the non-king card. So, the number of favorable outcomes for this scenario is (4 * 48).

Finally, we can sum up the number of favorable outcomes for each scenario: (4 * (48^3)) + (6 * (48^2)) + (4 * 48). This gives us the total number of favorable outcomes.

Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:

Probability = (Total favorable outcomes) / (Total possible outcomes)

I hope you enjoy calculating that! Remember, if you run into any "joker" cards in your deck, just toss them aside. Good luck!

To find the probability of drawing at least 1 king, we can use the principle of complementary probability.

Step 1: Determine the total number of possible outcomes.
We start by finding the total number of ways to draw 4 cards from a standard deck of 52 cards. This can be calculated using the combination formula:
C(52, 4) = 52! / (4!(52-4)!) = 270,725

Step 2: Determine the number of outcomes without any kings.
The number of ways to draw 4 cards without any kings can be calculated as follows:
Number of non-king cards = 52 - 4 = 48
Number of ways to draw 4 non-king cards = C(48, 4) = 48! / (4!(48-4)!) = 194,580

Step 3: Calculate the probability of drawing at least 1 king.
The probability of drawing at least 1 king can be determined by subtracting the probability of not drawing any kings from 1:
P(at least 1 king) = 1 - P(no kings)
P(no kings) = (Number of ways to draw 4 non-king cards) / (Total number of possible outcomes)
= 194,580 / 270,725
≈ 0.718

P(at least 1 king) = 1 - 0.718
≈ 0.282

Therefore, the probability of drawing at least 1 king from the game is approximately 0.282 or 28.2%.

To find the probability of drawing at least one king from a deck of cards, we need to determine the number of favorable outcomes and the total number of possible outcomes.

First, let's calculate the total number of possible outcomes. Since we are drawing 4 cards from an ordinary deck, there are 52 cards to choose from for each draw. So, for each card, we have 52 options. Therefore, the total number of possible outcomes is 52^4.

Next, let's determine the number of favorable outcomes, which represents the situations where at least one king is drawn. There are two scenarios to consider:

1. Drawing exactly one king:
- There are 4 kings in the deck, so we can choose 1 king in C(4,1) = 4 ways.
- For the remaining 3 cards, we can choose them from the remaining 48 non-king cards in C(48,3) ways.
- Therefore, the number of favorable outcomes for drawing exactly one king is 4 * C(48,3).

2. Drawing more than one king:
- There are 4 kings in the deck, so we can choose 2, 3, or 4 kings in C(4,2), C(4,3), and C(4,4) ways respectively.
- For the remaining 2, 1, or 0 cards, we can choose them from the remaining 48 non-king cards in C(48,2), C(48,1), and C(48,0) ways respectively.
- Therefore, the number of favorable outcomes for drawing more than one king is C(4,2) * C(48,2) + C(4,3) * C(48,1) + C(4,4) * C(48,0).

Finally, to calculate the probability of drawing at least one king, we divide the number of favorable outcomes by the total number of possible outcomes:

P(at least one king) = (number of favorable outcomes) / (total number of possible outcomes)

P(at least one king) = [4 * C(48,3) + C(4,2) * C(48,2) + C(4,3) * C(48,1) + C(4,4) * C(48,0)] / (52^4)

After plugging in the values and performing the calculations, you will obtain the probability of drawing at least one king from the given scenario.