Question

Choosing a Mate Men and women often disagree on how they think about selecting a mate. Suppose that a poll of 1000 individuals in their twenties gave the following responses to the question of whether it is more important for their future mate to be able to communicate their feelings ( \(F\) ) than it is for that person to make a good living \((G)\). $$\begin{array}{lccc} & \text { Feelings }(F) & \text { Good Living }(G) & \text { Totals } \\\\\hline \text { Men }(M) & .35 & .20 & .55 \\\\\text { Women }(W) & .36 & .09 & .45 \\\\\hline \text { Totals } & 71 & 29 & 1.00\end{array}$$ If an individual is selected at random from this group of 1000 individuals, calculate the following probabilities: a. \(P(F)\) b. \(P(G)\) c. \(P(F \mid M)\) d. \(P(F \mid W)\) e. \(P(M \mid F)\) f. \(P(W \mid G)\)

Short answer

Answer: a. The probability of communication of feelings being the most important quality (P(F)) is 0.71. b. The probability of making a good living being the most important quality (P(G)) is 0.29. c. The probability of a man choosing feelings over a good living (P(F|M)) is 0.35. d. The probability of a woman choosing feelings over a good living (P(F|W)) is 0.36. e. The probability of a man's response given the importance of feelings (P(M|F)) is approximately 0.493. f. The probability of a woman's response given the importance of a good living (P(W|G)) is approximately 0.310.

Step-by-step solution

(a) Calculating P(F)

To calculate the probability that communication of feelings is the most important quality (P(F)), we notice that it is provided as a row total in the table given. P(F) = 0.71.

(b) Calculating P(G)

Similarly, to calculate the probability that making a good living is the most important quality (P(G)), we look for the row total in the table. P(G) = 0.29.

(c) Calculating P(F|M)

The probability of a man choosing feelings over a good living, P(F|M), can be calculated directly from the table as the intersection of 'Men' and 'Feelings'. P(F|M) = 0.35.

(d) Calculating P(F|W)

Similarly, the probability of a woman choosing feelings over a good living, P(F|W), can be calculated directly from the table as the intersection of 'Women' and 'Feelings'. P(F|W) = 0.36.

(e) Calculating P(M|F)

To calculate the probability of a man's response given the importance of feelings, P(M|F), we use the conditional probability formula: P(M|F) = P(M∩F) / P(F) From the table, we find P(M∩F) = 0.35 and P(F) = 0.71, So, P(M|F) = 0.35 / 0.71 ≈ 0.493.

(f) Calculating P(W|G)

To calculate the probability of a woman's response given the importance of a good living, P(W|G), we use the conditional probability formula again: P(W|G) = P(W∩G) / P(G) From the table, we find P(W∩G) = 0.09 and P(G) = 0.29, So, P(W|G) = 0.09 / 0.29 ≈ 0.310. Thus, the probabilities we were asked to calculate are: a. P(F) = 0.71 b. P(G) = 0.29 c. P(F|M) = 0.35 d. P(F|W) = 0.36 e. P(M|F) ≈ 0.493 f. P(W|G) ≈ 0.310

Key concepts

Probability Distribution

When we discuss probability distribution in the context of polling data, we are referring to how the probabilities of different outcomes are spread out across the possible events. In the exercise, we look at a survey of individuals answering questions about their preferences. This creates a probability distribution over the outcomes;

• The probability that an individual values communication of feelings most, denoted as \(P(F)\).
• The probability that an individual thinks making a good living is more important, \(P(G)\).

In the exercise, the totals given in the table (0.71 and 0.29) represent the probabilities for each option across the entire sample. This probability distribution helps us understand how likely each preference is within the group surveyed.

Probability Calculation

Calculating probabilities involves understanding both simple probabilities and conditional probabilities. In the exercise, some probabilities are straightforward, like \(P(F)=0.71\) and \(P(G)=0.29\), which refer to the overall likelihood of choosing feelings or a good living respectively.
Other probabilities involve conditions, such as \(P(F\mid M)\). Conditional probability looks at the likelihood of an event occurring given that another event has already occurred. For \(P(F\mid M)\), we find the chance a man prioritizes feelings. This is calculated directly from the proportions detailed in the table using the matching category data that intersects with the question at hand.
Remember, conditional probability uses the formula:\[P(A\mid B) = \frac{P(A \cap B)}{P(B)}\]where \(A\) is the desired event and \(B\) is the condition. This formula captures the idea of narrowing down our sample space to just those who meet the condition \(B\), and finding \(A\) within that group.

Polling Data Analysis

Polling data analysis, particularly in the context of the exercise about mate selections, involves understanding what the raw numbers and probabilities tell us about people's preferences.
Firstly, it is crucial to break down the results by different groups. For instance, analyzing responses from men and women separately using conditional probabilities like \(P(F\mid M)\) and \(P(F\mid W)\). Such analysis reveals whether there are significant differences in priority based on gender.
Additionally, using conditional probabilities such as \(P(M\mid F)\) or \(P(W\mid G)\), helps us interpret results like what proportion of those valuing feelings are men, or what proportion of those valuing a good living are women.

• This kind of analysis helps identify trends or patterns among sub-groups within the dataset.
• It provides deeper insights beyond the surface level raw probabilities, contributing to more informed conclusions about the surveyed population.