4.7 Discrete Distribution (Playing Card Experiment)

Discrete Probability Distributions

Playing Card Probability Experiment

A playing card experiment uses a standard 52-card deck (4 suits: hearts, diamonds, clubs, spades; 13 ranks each) to illustrate how discrete probability distributions work in practice. By drawing cards and recording outcomes, you can see how theoretical probabilities compare to what actually happens.

The random variable $X$ represents the number of "successes" across your draws, where you define what counts as a success beforehand (e.g., drawing a face card, drawing a red ace). Because $X$ can only take whole-number values (0, 1, 2, ...), this is a discrete probability distribution.

Here's how to set up the experiment:

1. Define your success criteria (e.g., "drawing a king" or "drawing a red card").
2. Decide whether you're drawing with or without replacement (this matters for probability calculations).
3. Perform the experiment over many trials, recording the number of successes each time.
4. Calculate the relative frequency of each outcome by dividing the count of that outcome by the total number of trials.

As you increase the number of trials, your relative frequencies will get closer to the theoretical probabilities. This is the law of large numbers in action. Ten trials might give you results that look nothing like the theory, but by 100 or 500 trials, the pattern tightens up.

One detail to watch: if you draw with replacement (put the card back and shuffle before the next draw), each draw is independent and the probability stays constant at $1/52$ for any specific card. Without replacement, the draws are not independent, because the deck composition changes after each draw.

Multiplication Rule for Card Combinations

The multiplication rule says that the probability of two independent events both occurring equals the product of their individual probabilities. This is how you calculate the probability of drawing specific card combinations.

With replacement (independent draws):

Each draw has the same probabilities because the card goes back into the deck.

• Probability of drawing two kings in a row: $P = \frac{4}{52} \times \frac{4}{52} = \frac{16}{2704} \approx 0.0059$

There are 4 kings in the deck on both draws, so the probability stays the same.

Without replacement (dependent draws):

The deck shrinks by one card after each draw, and the number of "target" cards may also decrease.

• Probability of drawing two kings in a row: $P = \frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} \approx 0.0045$

After you draw the first king, only 3 kings remain in a 51-card deck. Notice the probability is lower than the with-replacement case.

• Probability of drawing the king of hearts followed by the queen of spades: $P = \frac{1}{52} \times \frac{1}{51} = \frac{1}{2652} \approx 0.000377$

The queen of spades is still in the deck after you remove the king of hearts, so the numerator stays 1, but the deck size drops to 51.

Without-replacement scenarios can also be modeled using the hypergeometric distribution, which you may encounter later in the course.

Expected Values and Variance Interpretation

The expected value (mean) of a discrete distribution tells you the long-run average outcome. It's calculated by multiplying each possible value of $X$ by its probability, then summing:

$E(X) = \sum_{i=1}^{n} x_i \cdot P(X = x_i)$

For example, if $X$ counts the number of red cards drawn in 5 draws (with replacement), the probability of drawing a red card on any single draw is $26/52 = 0.5$. The expected value would be $E(X) = 5 \times 0.5 = 2.5$. You won't draw exactly 2.5 red cards in any single trial, but over many trials, the average will settle near 2.5.

Variance measures how spread out the outcomes are around the expected value:

$Var(X) = \sum_{i=1}^{n} (x_i - E(X))^2 \cdot P(X = x_i)$

A higher variance means results are more scattered. A lower variance means outcomes cluster tightly around the mean.

When you run your experiment, compare your results to these theoretical values:

• If your experimental mean is close to $E(X)$, the experiment is behaving as the theory predicts.
• If there's a noticeable gap, it's likely due to random variation, especially with a small number of trials. More trials will typically bring the experimental values closer to the theoretical ones.

Probability Functions and Distributions

Two functions describe discrete distributions:

• The probability mass function (PMF) gives the probability that $X$ equals each specific value. For example, $P(X = 2) = 0.25$ means there's a 25% chance of getting exactly 2 successes.
• The cumulative distribution function (CDF) gives the probability that $X$ is less than or equal to a specific value. So $P(X \leq 2)$ adds up the probabilities for $X = 0$, $X = 1$, and $X = 2$.

Two common discrete distributions connect directly to card experiments:

• The Bernoulli distribution models a single trial with two outcomes: success or failure. Drawing one card and checking whether it's a heart (success) or not (failure) is a Bernoulli trial.
• The binomial distribution extends this to multiple independent trials. If you draw a card with replacement 10 times and count how many hearts you get, that count follows a binomial distribution with $n = 10$ and $p = 13/52 = 0.25$.