Question

A woman brought a complaint of gender discrimination to an eight-member human relations advisory board. The board, composed of five women and three men, voted \(5-3\) in favor of the plaintiff, the five women voting for the plaintiff and the three men against. Has the board been affected by gender bias? That is, if the vote in favor of the plaintiff was \(5-3\) and the board members were not biased by gender, what is the probability that the vote would split along gender lines (five women for, three men against)?

Short answer

Answer: The probability is \(\frac{7}{32}\).

Step-by-step solution

Determine the size of the sample space

Since there are 8 members on the board, each with 2 options (vote for or against), there are 2^8 possible vote combinations.

Define a vote combination

A vote combination is defined as the number of ways the board could vote for the plaintiff. For example, the given vote combination is 5 votes in favor (five women) and 3 votes against (three men).

Calculate the binomial coefficient for the given vote combination

We can use the binomial coefficient formula to find the number of ways the given vote combination of 5 votes in favor (from a total of 8 members) can occur: Number of Ways = \(\binom{8}{5}\) = \(\frac{8!}{5!(8-5)!}\) = 56 ways

Determine the probability of the given vote combination

Now we need to find the probability of the given vote combination happening if the members were not biased by gender. Since we assume that each member can either vote in favor or against with equal probability (not biased), the probability of the given vote combination would be the number of ways that combination can happen divided by the total number of possible vote combinations: Probability of given vote combination = \(\frac{Number\:of\:Ways}{Total\:number\:of\:possible\:vote\:combinations}\) = \(\frac{56}{2^8}\) = \(\frac{56}{256}\) = \(\frac{7}{32}\) The probability that the vote would split along gender lines (five women for, three men against) if the board members were not biased by gender is \(\frac{7}{32}\).

Key concepts

Binomial Coefficient

The binomial coefficient is a mathematical tool used to count how many ways you can choose items from a larger pool without caring about the order of those items. In our exercise, the coefficient helps determine how many different ways five people can vote in favor from a total of eight members, which forms part of our analysis for a possible bias.

The formula to find a binomial coefficient is represented as \( \binom{n}{k} \) and is equal to \( \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of items to choose from, \( k \) is the number of items chosen, and \( ! \) denotes factorial, meaning the product of all positive integers up to that number.

In the exercise, using \( \binom{8}{5} \), you calculate the number of ways the board could have voted 5-3. After applying the formula, you find there are 56 ways of forming this particular voting combination. The binomial coefficient, thus, is crucial in establishing the number of possible unbiased vote outcomes.

Sample Space

The sample space in probability theory refers to all possible outcomes of an experiment or event. In our case, the "experiment" is the board members casting their votes, each having a choice to vote either for or against.

For this scenario with eight board members and two voting options for each ("for" or "against"), the sample space is vast. It includes all possible combinations of votes, totaling \( 2^8 \) or 256 unique possibilities. This comprehensive set of outcomes is essential as it represents the benchmark from which probabilities are derived.

Understanding the sample space allows us to appreciate the probability of any one specific voting outcome occurring, like a 5-3 split. This clarity is key to determining how likely it is that votes would naturally align along genders without bias.

Gender Bias

Gender bias occurs when an outcome is influenced by gender, rather than being a result of independent decision-making. Within the context of the exercise, we are questioning if the board's decision, with females favoring the plaintiff and males against, is due to gender bias.

To investigate this, we calculate the probability of the 5-3 vote split occurring naturally without bias. If the probability is particularly low, it might suggest bias; however, further analysis and context are necessary before reaching any conclusions. Calculating this probability aims to provide statistical evidence for or against the presence of gender bias in the board's decision-making process.

Interpreting these probabilities requires comprehensive understanding, as numerical results can highlight potential bias but do not confirm causation. Other factors could influence the decision beyond gender alignment.

Voting Probability

Voting probability examines the chance of various outcomes in a given voting situation, especially when assuming unbiased conditions. In the given exercise, calculating the probability of the 5-3 split without gender influence is central.

With objective factors considered, such as each member having an independent and equal likelihood of voting either way, the calculated probability of the 5-3 gender-aligned split is \( \frac{7}{32} \) or approximately 21.875%.

This percentage demonstrates a certain likelihood of this outcome occurring under the assumption of impartial voting. It serves as a benchmark to compare against potential bias indicators and understand the fairness of the voting process. Understanding voting probability allows analysts to differentiate between expected statistical outcomes and outliers that might suggest underlying biases or other influential factors.