Start with a standard, uniformly shuffled 52-card deck, and draw the cards one at a time. What is the probability of drawing a King strictly before getting a Spade?
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To calculate the probability of drawing a King strictly before getting a Spade, we need to determine the number of desired outcomes (favorable outcomes) and the total number of possible outcomes.
Let's break down the problem step by step:
Step 1: Determine the desired outcomes.
To find the desired outcomes, we need to consider the order of the cards drawn. In this case, we want to draw a King before drawing any Spades. Therefore, the desired outcomes are the scenarios where we draw a King before drawing any Spades.
Step 2: Determine the total number of outcomes.
In a standard 52-card deck, the total number of outcomes is 52!, which represents the number of ways to arrange all the cards in the deck.
Step 3: Calculate the probability.
The probability is calculated by dividing the number of desired outcomes by the total number of outcomes.
To determine the number of desired outcomes, we can consider the two cases when drawing a King before getting a Spade:
Case 1: Drawing a King as the first card.
In this case, we have only one desired outcome since there are four Kings in the deck.
Case 2: Drawing a King after drawing some other cards but before getting any Spades.
To calculate the number of desired outcomes for this case, we need to consider the order of cards drawn. We have four Kings that can be placed in any of the first four positions, followed by a sequence of non-Spade cards that can be arranged in the remaining spots. The remaining cards in the deck are all Spades, which can be placed in the remaining positions. Therefore, the number of desired outcomes for this case is 4 * (48!) * (4!).
The total number of desired outcomes is the sum of the two cases.
Finally, we calculate the probability using the formula:
Probability = Number of desired outcomes / Total number of outcomes
By plugging in the values from the previous steps, you can calculate the probability of drawing a King strictly before getting a Spade from a standard, uniformly shuffled 52-card deck.