Calculating the Probability of Drawing Specific Cards in Multiple Draws with Replacement
Understanding the probability of drawing specific cards in a deck, especially when drawing multiple times with replacement, is a common question in both gaming and mathematical probability theory. This article will explore the scenario of drawing from a 50-card deck, with a focus on identifying the probability of seeing at least one of eight specific cards in two separate draws, each consisting of five cards, with reshuffling between draws.
Step-by-Step Approach to Calculating the Probability
The problem at hand involves a deck of 50 cards containing 8 copies of a specific card. We want to find the probability of drawing at least one of those 8 cards in two separate draws of 5 cards each, with reshuffling between the draws.
1. Probability of Not Drawing the Specific Card in One Draw
The deck contains 42 non-target cards and 8 target cards. To find the probability of not drawing a target card in one draw of 5 cards, we consider the following steps:
The probability of drawing 5 cards from the 42 non-target cards initially is calculated as:P(text{no target card in 5 cards}) frac{42}{50} times frac{41}{49} times frac{40}{48} times frac{39}{47} times frac{38}{46}
This simplifies to approximately:P(text{no target card in 5 cards}) approx 0.630
2. Probability of Not Drawing the Card in Both Draws
Since the cards are reshuffled between draws, each draw is independent. Therefore, the probability of not drawing the target card in both draws is the square of the probability of not drawing it in one draw:
P(text{no target card in both draws}) (0.630)^2 approx 0.397
3. Probability of Drawing the Card at Least Once
The probability of drawing at least one of the 8 target cards in the two draws is the complement of the probability of not drawing it in both draws:
P(text{at least one target card}) 1 - P(text{no target card in both draws}) 1 - 0.397 0.603
Thus, the probability of seeing at least one of the 8 cards in two 5-card draws, with reshuffling in between, is approximately 60.3%.
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