Problem 42 - A First Course in Probability

Question

A group of 20 people consisting of 10 men and 10 women is randomly arranged into 10 pairs of 2 each. Compute the expectation and variance of the number of pairs that consist of a man and a woman. Now suppose the 20 people consist of 10 married couples. Compute the mean and variance of the number of married couples that are paired together.

Step-by-Step Solution

Answer

The expectation and variance for the number of mixed-gender pairs are given by: \( E(mixed) = 10 * P(mixed) \) \( Var(mixed) = E(X^2) - [E(X)]^2 \) For the number of married couples paired together, the expectation and variance are given by: \( E(married) = 10 * P(married) \) \( Var(married) = E(Y^2) - [E(Y)]^2 \)

Step 1: Calculate the Total Number of Pairing Possibilities

First, let's find the total possible pairings that can be formed with 20 people (10 men and 10 women). Using the combination formula, the total number of possible pairs is given by the expression: \( C(20, 2) = \frac{20!}{(20-2)!2!} \)

Step 2: Calculate the Total Number of Mixed-Gender Pairing Possibilities

Now, let's find the total number of mixed-gender pairs. We have 10 men and 10 women, so the total number of mixed-gender pairs is: \( 10 * 10 = 100 \)

Step 3: Calculate the Probability of a Mixed-Gender Pair

Next, let's calculate the probability that a pair is mixed gender. To do this, divide the number of mixed-gender pairings (found in step 2) by the total number of pairing possibilities (found in step 1): \( P(mixed) = \frac{100}{C(20, 2)} \)

Step 4: Calculate the Expectation

To find the expectation, we can multiply the probability of getting a mixed pair by the total number of pairs: \( E(mixed) = 10 * P(mixed) \)

Step 5: Calculate the Variance

To find the variance, we can use the formula: \( Var(mixed) = E(X^2) - [E(X)]^2 \) where X represents the random variable for the number of mixed-gender pairs. #--Scenario 2: Expectation and Variance of Married Couples Paired Together --#

Step 6: Calculate the Total Number of Ways to Pair the Couples

Now, let's find the total number of ways to pair the 10 married couples. For this, we will use the combination formula again: \( C(20, 2) * C(18, 2) * ... * C(4, 2) * C(2, 2) \)

Step 7: Calculate the Total Number of Ways to Pair Married Couples Together

We can compute the total number of ways to arrange the 10 married couples using the expression: \( 10! \)

Step 8: Calculate the Probability of a Married Couple Being Paired Together

Divide the total number of ways to pair married couples together (found in step 7) by the number of ways to pair the couples (found in step 6): \( P(married) = \frac{10!}{C(20, 2) * C(18, 2) * ... * C(4, 2) * C(2, 2)} \)

Step 9: Calculate the Expectation and Variance

Finally, we will calculate the expectation and variance in the same manner as we did for the mixed-gender pairs. \( E(married) = 10 * P(married) \) \( Var(married) = E(Y^2) - [E(Y)]^2 \)

Key Concepts

Expectation

In probability theory, the term "expectation" refers to the average or "expected" value of a random variable. It's essentially what we anticipate happening on average in a random experiment. Imagine you have a dice, and you roll it many times; the average outcome you get is the expectation. In the context of pairing people in our problem, we are calculating the typical number of mixed-gender or married couple pairs.

To compute the expectation, we use probabilities. For mixed-gender pairs, we identify the chance of selecting a man and a woman as a pair. We find this probability and use it to find the expected number of such pairs when everyone is paired. In our exercise, multiplying the probability by the total number of pairs gives us the expectation. This approach applies similarly when calculating the expectation of married couples being paired together.

Thus, expectation helps us predict typical outcomes in chance-based scenarios, whether it's mixing genders in pairs or keeping married couples together.

Variance

Variance is another crucial concept in probability theory that measures how much variability there is from the expectation. In simpler terms, it tells us how "spread out" the possible outcomes are around the expectation. A low variance indicates that the outcomes are generally close to the expected value, while a high variance signifies a wide spread of possible results.

To find variance in this exercise, we need to calculate the expected value of the square of the random variable, and then subtract the square of the expectation. Mathematically, it's expressed as:
\[ Var(X) = E(X^2) - [E(X)]^2 \]

This calculation helps us understand how many pairs deviate from the expected number, whether it’s mixed-gender pairs or married couples. By computing the variance, we gain insights into the unpredictability surrounding our expected outcomes.

Combinatorics

Combinatorics is the mathematical study of counting, arrangement, and combination of objects. It plays a vital role in problems involving probabilities about how items can be grouped. In our scenario, we use combinatorics to calculate how people can be arranged into pairs in various ways.

The combination formula, denoted as \( C(n, k) \), represents the number of ways to choose \( k \) items from \( n \) items without regard to the order. For example, \( C(20, 2) \) gives the number of ways to create a single pair from 20 people.

Combinatorics provides us the tools to understand and calculate different possible configurations of pairs, whether we're looking at any two people, mixed-gender partners, or married couples pairing together. By understanding the number of possible arrangements, we can then compute the probabilities and expectations effectively.

Mixed-Gender Pairs

Creating pairs from a group that contain both a man and a woman forms what we call mixed-gender pairs. This exercise requires calculating the number of such pairs possible and their likelihood in a random pairing scenario.

With 10 men and 10 women, envision having each man paired with each woman. This gives us a basic idea of mixed-gender pairings, calculated simply by multiplying the number of men by the number of women, which totals to 100 mixed-gender pair possibilities.

However, not all pairings will result in mixed-gender groups, so determining the probability that a random pair is mixed-gender involves dividing these possibilities by the total possible number of pairings. This probability helps to calculate the expected number of mixed-gender pairs when the group is fully paired.

Married Couples Probability

Calculating the probability that married couples are paired together in random pairings poses a fascinating problem. With 10 married couples involved, we need to determine how likely it is for each to end up paired with their spouse.

Using combinatorial methods, we calculate all the ways to arrange pairs, then the specific arrangements that keep each couple together. The probability of married couples being paired involves the ratio of these special pairings to all possible pairings.

To determine the expectation, we multiply the probability by the total possible pairs, offering insights into how common it is for couples not to be separated. Variance then accompanies this calculation, indicating how often actual pairings diverge from our expected scenario. These calculations not only show the likelihood but also the stability of keeping married couples together by chance.