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. (b) how many plants were tall and had peas that were neither smooth nor green? (c) how many plants were not tall but had peas that were smooth and green? 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. We know \(n(T) = 31\), \(n(G)=31\), \(n(S) = 44\), \(n(T\cap G)=13\), \(n(G\cap S)=22\), \(n(T\cap G\cap S)=7\), and the total number of non - empty plants \(n = 63 - 3=60\).

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

By the principle of inclusion - exclusion \(n(T\cup G\cup S)=n(T)+n(G)+n(S)-n(T\cap G)-n(T\cap S)-n(G\cap S)+n(T\cap G\cap S)\). Since \(n(T\cup G\cup S) = 60\), we can solve for \(n(T\cap S)\).
\[60=31 + 31+44-13 - n(T\cap S)-22 + 7\]
\[60=78 - n(T\cap S)\]
\[n(T\cap S)=18\]

Step3: For part (b)

The number of plants that are tall and have non - green and non - smooth peas:
We know that the number of plants that are tall and have either green or smooth or both is \(n((T\cap(G\cup S)))\). First, \(n(G\cup S)=n(G)+n(S)-n(G\cap S)=31 + 44-22 = 53\). Then \(n(T\cap(G\cup S))=n(T)+n(G\cup S)-n(T\cup G\cup S)\). But we can also use the fact that the number of plants that are tall and have non - green and non - smooth peas is \(n(T)-n(T\cap(G\cup S))\).
We know that \(n(T\cap(G\cup S))=n((T\cap G)\cup(T\cap S))=n(T\cap G)+n(T\cap S)-n(T\cap G\cap S)=13 + 18-7=24\). So the number of plants that are tall and have non - green and non - smooth peas is \(n(T)-n((T\cap G)\cup(T\cap S))=31 - 24 = 7\).

Step4: For part (c)

The number of plants that are not tall but have smooth and green peas is \(n(G\cap S)-n(T\cap G\cap S)=22 - 7=15\).

Answer:

15