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White Cats and Deafness. Although cats generally possess an acute sense of hearing, due to an anomaly in their genetic makeup, deafness among white cats with blue eyes is quite common. Approximately \(95 \%\) of the general cat population are non-white cats (i.e., not pure white), and congenital deafness is extremely rare in non-white cats. However, among white cats, approximately \(75 \%\) with two blue eyes are deaf, \(40 \%\) with one blue eye are deaf, and only \(19 \%\) with eyes of other colors are deaf. Additionally, among white cats, approximately \(23 \%\) have two blue eyes, \(4 \%\) have one blue eye, and the remainder have eyes of other colors. 9 (a) Draw a tree diagram for selecting a white cat (outcomes: one blue eye, two blue eyes, or eyes of other colors) and deafness (outcomes: deaf or not deaf). (b) What is the probability that a randomly chosen white cat is deaf?

Step by step solution

01

Define Categories

We have categories for white cats based on eye color: (1) two blue eyes, (2) one blue eye, and (3) other colored eyes. Additionally, for deafness, we have cats that are either (a) deaf or (b) not deaf.

02

Calculate Probabilities Based on Eye Color

From the problem, we have: - Two blue eyes: 23% - One blue eye: 4% - Eyes of other colors: 100% - 23% - 4% = 73%

03

Probabilities of Deafness for Each Eye Color

Based on the information: - Two blue eyes, deaf: 75% - One blue eye, deaf: 40% - Other colored eyes, deaf: 19%

04

Calculate Probability of Deafness for Each Eye Category

We multiply the probability of having a certain eye color by the probability of being deaf given that eye color: - Two blue eyes: 0.23 * 0.75 = 0.1725 - One blue eye: 0.04 * 0.40 = 0.016 - Eyes of other colors: 0.73 * 0.19 = 0.1387

05

Calculate Total Probability of Deafness

Add the probabilities from each category to find the total probability that a randomly chosen white cat is deaf:\[ P(\text{deaf}) = 0.1725 + 0.016 + 0.1387 = 0.3272 \]

06

Highlight Final Answer

The probability that a randomly chosen white cat is deaf is approximately 32.72%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tree Diagram

A tree diagram is a valuable tool in probability to visually map out possible outcomes of a problem. It starts with a single node that branches out into several nodes, illustrating all possible scenarios. In the context of the white cats and deafness problem, the first level of the tree diagram represents different eye color outcomes for white cats:

• Two blue eyes
• One blue eye
• Other colored eyes

From each of these nodes, branches split further to show the probabilities of either being deaf or not deaf. This branching helps keep track of all combinations of eye color and deafness:

• For white cats with two blue eyes: 75% are deaf
• For white cats with one blue eye: 40% are deaf
• For white cats with other colored eyes: 19% are deaf

Using a tree diagram allows for a clear and systematic approach to understanding all possible outcomes and their associated probabilities.

Conditional Probability

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. In the case of white cats and deafness, conditional probability helps determine the likelihood of deafness based on the eye color of the cats. If you know that a particular white cat has one blue eye, the probability that this cat is deaf is 40%. This value is derived from the information given in the exercise, which categorizes cats based on their eye color. Conditional probability is calculated by multiplying the probability of having a particular eye color by the probability of deafness given that eye color. For example:

• Probability of a white cat having two blue eyes and being deaf is 23% \( \times \) 75% = 17.25%
• Probability of a white cat with one blue eye being deaf is 4% \( \times \) 40% = 1.6%

This illustrates the necessity of understanding how prior conditions, like eye color, affect the probability of deafness in white cats.

Genetic Anomalies

Genetic anomalies refer to irregularities in the genetic code which can lead to observable physical traits or health conditions in organisms. In cats, such anomalies manifest notably in the form of congenital deafness, particularly among white cats with certain eye colors. The anomaly in concern here involves the relationship between coat color, eye color, and hearing ability. Among the genetic makeup of white cats:

• A large proportion with two blue eyes (75%) experience deafness.
• Those with one blue eye show a lower chance (40%) of being deaf.
• Cats with eyes of other colors have even lower associated deafness rates (19%).

These statistics illustrate how specific genetic traits can influence certain health outcomes. Genetics plays a key role in such probability problems as it determines base conditions that significantly affect outcomes.

Statistics Problem Solving

Solving statistics problems often involves using structured methods to break down complex information into manageable parts. In this exercise, we have intertwined variables: cat color and the probability of deafness. By focusing on the eye color of white cats, you can calculate the overall probability of a white cat being deaf. The resolution follows these steps:

• Identify relevant categories based on known attributes (e.g., eye color in cats).
• Assign probabilities to each category's potential outcome (e.g., deaf or not deaf).
• Combine individual probabilities to derive a total probability of interest. As calculated, the probability of any given white cat being deaf is 32.72%.

This demonstrates the importance of a methodical approach in statistics, facilitating clear calculation paths from raw data to meaningful conclusions.