Problem 11 - Algebra 2

Question

Make a tree diagram based on the survey results below. Then find \(P(\text { a female }\) respondent is left-handed) and \(P(\text { a respondent is both male and right-handed). }\) \(\bullet\) Of all the respondents, 17\(\%\) are male. \(\bullet\) Of the male respondents, 33\(\%\) are left-handed. \(\bullet\) Of female respondents, 90\(\%\) are right-handed.

Step-by-Step Solution

Answer

The probability that a respondent is female and left-handed is 0.083 and that a respondent is male and right-handed is 0.1139.

Step 1: Draw Tree Diagram

Start by drawing a tree diagram using the given survey results. From the root of the tree, draw two branches, one to represent males(17\% of total population) and the other to represent females(83% of total population) because they sum up to 100%. Then, for each gender, draw two more branches: for males, one branch for being left-handed (33%) and the other for being right-handed (67%). For females, one branch for being left-handed (10%) and the other for being right-handed (90%).

Step 2: Compute P(female and left-handed)

The probability that a respondent is female and left-handed is the product of the probabilities along the path from the root of the tree to the 'female and left-handed' node. So, P(female and left-handed) = P(female) * P(left-handed | female) = 0.83 * 0.1 = 0.083.

Step 3: Compute P(male and right-handed)

Similarly, the probability that a respondent is male and right-handed is determined by multiplying the probabilities along the path from the root to the 'male and right-handed' node. So, P(male and right-handed) = P(male) * P(right-handed | male) = 0.17 * 0.67 = 0.1139.

Key Concepts

Tree Diagram

A tree diagram is a visual representation that helps break down complex probability problems into simpler parts. When faced with a problem involving multiple categories and subcategories, a tree diagram is an ideal tool.

In the given exercise, the tree diagram starts with two main branches representing the two genders surveyed: male and female. Since 17% of the respondents are male, we draw a branch for males reflecting this percentage. The remaining 83% goes into a separate branch for females. Each gender then has sub-branches based on their handedness—left-handed or right-handed.

• For males, 33% are left-handed and thus 67% are right-handed.
• For females, 10% are left-handed and 90% are right-handed.

This branching continues based on the given probabilities. By visually mapping out the possibilities, a tree diagram helps ensure all potential outcomes are accounted for and simplifies the process of calculating specific combined probabilities later on.

Conditional Probability

Conditional Probability is the likelihood of an event occurring given that another event has already taken place. It is denoted by the symbol \(P(B|A)\), which reads 'the probability of B given A'.

In our exercise, we're using conditional probability to find two aspects: the probability of a female being left-handed, and the probability of a male being right-handed. Using the tree diagram:

• The probability of selecting a female who is left-handed is determined by multiplying the probability of choosing a female by the probability of a female being left-handed: \(P( ext{female and left-handed}) = P( ext{female}) \times P( ext{left-handed | female}) = 0.83 \times 0.1 = 0.083\).
• Similarly, for a male being right-handed: \(P( ext{male and right-handed}) = P( ext{male}) \times P( ext{right-handed | male}) = 0.17 \times 0.67 = 0.1139\).

By using the conditional probabilities, you can efficiently narrow down possibilities to calculate the probability of overlapping events within a total sample space.

Survey Analysis

Survey Analysis involves interpreting data gathered from questionnaires or polls to extract meaningful information. In this exercise, a survey collected data on gender and handedness, which are two categorical variables.

To perform effective survey analysis, consider these steps:

• Start by accurately categorizing the data, such as separating male and female responses.
• Within these categories, identify sub-groups and calculate their probabilities, like calculating the percentages of left-handed and right-handed individuals.
• Use tools like the tree diagram to visually map out the relationships and probabilities between different categories and subcategories.

Through this process, survey analysis facilitates an understanding of the underlying trends and patterns in the data, such as the prevalence of right or left-handedness among different genders. This understanding enables better decision-making based on the surveyed information.