Question

after a genetics experiment on 63 pea plants, the number of plants having certain characteristics was tallied, with the results given to the right. answer parts (a) through (c). (a) find the number of plants that were tall and had smooth peas. 18 plants (b) how many plants were tall and had peas that were neither smooth nor green? plants 31 were tall; 31 had green peas; 44 had smooth peas; 13 were tall and had green peas; 22 had green peas and smooth peas; 7 had all three characteristics; 3 had none of the characteristics.

Explanation:

Step1: Use the principle of inclusion - exclusion

Let \(T\) be the set of tall plants, \(G\) be the set of plants with green peas, and \(S\) be the set of plants with smooth peas. The total number of non - empty plants is \(n = 63-3=60\).
We know that \(n(T) = 31\), \(n(G)=31\), \(n(S) = 44\), \(n(T\cap G)=13\), \(n(G\cap S)=22\), \(n(T\cap G\cap S)=7\), and \(n = n(T)+n(G)+n(S)-n(T\cap G)-n(T\cap S)-n(G\cap S)+n(T\cap G\cap S)\).

Step2: First, find \(n(T\cap S)\)

We know that \(60=31 + 31+44-13 - n(T\cap S)-22 + 7\).
Simplify the right - hand side: \(31+31 + 44-13-22 + 7=78\).
So, \(60=78 - n(T\cap S)\).
Solve for \(n(T\cap S)\): \(n(T\cap S)=78 - 60=18\).

Step3: For part (b), find the number of plants that are tall and have non - smooth and non - green peas

First, find the number of plants that are not smooth and not green.
The number of plants that are either smooth or green is \(n(S\cup G)=n(S)+n(G)-n(S\cap G)=44 + 31-22=53\).
The number of plants that are neither smooth nor green is \(60 - 53 = 7\).
We use the principle of inclusion - exclusion for the intersection of the set of tall plants and the set of non - smooth and non - green plants.
We know that \(n(T)=n((T\cap S)\cup(T\cap G)\cup(T\cap\overline{S}\cap\overline{G}))\).
We know \(n(T) = 31\), \(n(T\cap G)=13\), \(n(T\cap S)=18\), and \(n(T\cap G\cap S)=7\).
The number of plants that are tall and either have green or smooth peas is \(n((T\cap G)\cup(T\cap S))=n(T\cap G)+n(T\cap S)-n(T\cap G\cap S)=13 + 18-7=24\).
So the number of plants that are tall and have peas that are neither smooth nor green is \(31-24 = 7\).

Answer:

(a) 18
(b) 7