Let X denote the number of aces a poker player receives in a five-card hand. Find the pdf and graph the cdf for X.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![**Title: Analyzing the Probability Distribution of Aces in a Five-Card Poker Hand**
**Introduction:**
In this exercise, we aim to determine the probability distribution function (pdf) and graph the cumulative distribution function (cdf) for a random variable \( X \). Here, \( X \) represents the number of aces a poker player receives in a five-card hand.
**Problem Statement:**
Let \( X \) denote the number of aces a poker player receives in a five-card hand. Find the pdf and graph the cdf for \( X \).
**Solution Approach:**
1. **Identify the possible values of \( X \):**
- Since there are 4 aces in a standard deck of 52 cards, and we draw 5 cards, the number of aces \( X \) can be 0, 1, 2, 3, or 4.
2. **Calculate the Probability Distribution Function (pdf):**
- For each \( x \) (0 to 4), calculate the probability \( P(X = x) \) using combinations:
\[
P(X = x) = \frac{{\binom{4}{x} \times \binom{48}{5-x}}}{\binom{52}{5}}
\]
- Here, \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose \( k \) successes in \( n \) trials.
3. **Graph the Cumulative Distribution Function (cdf):**
- Construct the cdf \( F(x) = P(X \leq x) \) by accumulating the pdf values.
**Graphical Explanation:**
The graph of the cdf will be a step function that increases at each permissible value of \( x \), reflecting the cumulative probability up to that number of aces.
By comprehending the pdf and cdf of the number of aces in a poker hand, players and enthusiasts can better appreciate the probabilistic dynamics of card games, enhancing both strategic planning and analytical understanding.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67df2f2b-9148-4dd5-b0ae-87328374df62%2F06323457-73ba-4a46-a7b9-8e73ceec3f7d%2Fufihfad_processed.jpeg&w=3840&q=75)
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