Problem 4 - A First Course in Probability

Question

Five men and 5 women are ranked according to their scores on an examination. Assume that no two scores are alike and all \(10 !\) possible rankings are equally likely. Let \(X\) denote the highest ranking achieved by a woman. (For instance, \(X=1\) if the top-ranked person is female.) Find \(\mathrm{P}\\{\mathrm{X}=\mathrm{i}\\}\) \(i=1,2,3, \ldots, 8,9,10\)

Step-by-Step Solution

Answer

The probabilities for each i are: - P(X=1) = 0 - P(X=2) = 0.033 - P(X=3) = 0.15 - P(X=4) = 0.283 - P(X=5) = 0.283 - P(X=6) = 0 - P(X=7) = 0 - P(X=8) = 0 - P(X=9) = 0 - P(X=10) = 0 These probabilities are obtained by calculating \(\mathrm{P}(X=\mathrm{i}) = \frac{(5-i)!(i-1)!(6-i)!}{10!}\) for each i, where i ranges from 1 to 10.

Step 1: Determine the number of desired outcomes

For each i, we need to determine in how many ways can a woman be at the i-th rank. To count this, we can consider the men and women relative to that specific rank. There will be 5-i men with a better ranking (on positions 1 to i-1) and i-1 women with a worse ranking (position i+1 onwards).

Step 2: Calculate the ways men and women can be arranged

We can use permutations to determine the different ways the men and women can be arranged in their respective remaining positions. For example, - Permutations of the 5-i men in the i-1 spots higher than the i-th rank, - Permutations of the i-1 women in the i-1 spots higher than the i-th rank, - Permutations of the remaining 5-(i-1) = 6-i women in the remaining spots below the i-th rank.

Step 3: Multiply the individual permutations

Now, we multiply the individual permutations to get the total number of ways each gender can be arranged. - Permutations of the 5-i men in the i-1 spots: \((5-i)!\) - Permutations of the i-1 women in the i-1 spots: \((i-1)!\) - Permutations of the remaining 6-i women in the remaining spots: \((6-i)!\) Therefore, the total number of desired arrangements for a particular i is given by \((5-i)!(i-1)!(6-i)!\).

Step 4: Calculate probabilities

Now that we know the desired outcomes, we can calculate the probabilities for each i, denoted by P(X=i), by dividing the number of desired outcomes by the total number of possible outcomes (10!). Thus, for each i: \(\mathrm{P}(X=\mathrm{i}) = \frac{(5-i)!(i-1)!(6-i)!}{10!}\) Let's calculate the probabilities for i = 1, 2, 3, ..., 8, 9, 10: - P(X=1) = \(\frac{4!0!5!}{10!}\) = 0 - P(X=2) = \(\frac{3!1!4!}{10!}\) = 0.033 - P(X=3) = \(\frac{2!2!3!}{10!}\) = 0.15 - P(X=4) = \(\frac{1!3!2!}{10!}\) = 0.283 - P(X=5) = \(\frac{0!4!1!}{10!}\) = 0.283 - P(X=6) = 0 - P(X=7) = 0 - P(X=8) = 0 - P(X=9) = 0 - P(X=10) = 0 Thus, we have found the probabilities for each i. Note that the probabilities for i greater than 5 are zero, since we have more men left to place than available higher positions.

Key Concepts

Ranking Problems

In the context of ranking problems, we encounter scenarios where we need to determine the order or arrangement of participants based on specific criteria or scores. Here, we're specifically examining the rankings of five men and five women, each with unique scores. This involves assigning each individual a distinct rank, from 1 to 10, without any ties.

Ranking problems address the challenge of determining the probability that a particular person (or group, such as women) is at a specific rank. This is particularly interesting when the scores are not tied, leading to a unique sequence of rankings. This challenge requires understanding permutations and probability theory to accurately predict outcomes.

Permutations

Permutations play a crucial role in analyzing ranking problems. A permutation is an arrangement of all the members of a set into some sequence or order. For our problem, permutations help determine how to arrange men and women in different ranks.

When calculating permutations, we assess how many ways we can arrange a subset of individuals. For the given problem, permutations are used to determine different configurations of men and women relative to a specific rank. For instance:

• Permutations of the 5-i men who have ranks better than i
• Permutations of the i-1 women who have ranks worse than i
• Permutations of the remaining women for the ranks below i

Permutations allow us to mathematically represent the possible orders using factorial notation. For example, \[(n!)\] represents the total number of ways to arrange \( n \) individuals.

Probability Distributions

Probability distributions help describe how likely it is for different outcomes to occur within a given scenario. In this problem, we're interested in finding the probability \[(P(X=i)) \] where a woman is ranked as the highest at a specific rank \(i\).

The process involves determining the number of favorable outcomes where a woman holds the i-th rank while distributing the remaining individuals. By dividing the number of favorable permutations by the total possible permutations, we establish a probability distribution for each possible rank.

• The total possible arrangements are accounted for by \(10!\) which includes all possible sequences of 10 distinct ranks.
• The numerator of the probability formula takes into consideration permutations aligning with the desired condition (a woman at rank \(i\)).

This mathematical approach helps create a structured way to view and interpret the randomness involved in the rankings.

Combinatorial Analysis

Combinatorial analysis is the branch of mathematics that deals with counting, arrangement, and combination of objects. In the context of this exercise, it is essential for determining the number of ways to arrange distinct individuals into specified orders.

Combinatorial tools simplify complex ranking problems by breaking down each scenario into countable arrangements using permutations and factorials. This exercise utilizes combinatorial analysis by calculating:

• Potential positions for men and women before, at, and after the i-th position.
• Different ways to fill remaining ranking positions using permutations.

By using combinatorial analysis, we efficiently quantify large sets of possible arrangements, thereby aiding in the probability calculations.