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Round peas are dominant over recessive wrinkled peas. First, second, and third generations can be analyzed using these tables, where [tex]$R$[/tex] represents round and [tex]$r$[/tex] represents wrinkled. When second-generation round pea plants (each of which carries both [tex]$R$[/tex] and [tex]$r$[/tex] genes) are crossed, find the probability that a third-generation offspring will have (a) round peas and (b) wrinkled peas.

(a) The probability that a third-generation offspring has round peas is [tex]$\square$[/tex] [tex]$\square$[/tex].

First to Second Generation
\begin{tabular}{|l|c|c|}
\cline { 2 - 3 }
\multicolumn{1}{c|}{} & \multicolumn{2}{|c|}{Second Parent} \\
\hline
First Parent & [tex]$r$[/tex] & [tex]$r$[/tex] \\
\hline
[tex]$R$[/tex] & [tex]$Rr$[/tex] & [tex]$Rr$[/tex] \\
\hline
[tex]$R$[/tex] & [tex]$Rr$[/tex] & [tex]$Rr$[/tex] \\
\hline
\end{tabular}

Second to Third Generation
\begin{tabular}{|l|c|c|}
\cline { 2 - 3 }
\multicolumn{1}{c|}{} & \multicolumn{2}{|c|}{Second Parent} \\
\hline
First Parent & [tex]$R$[/tex] & [tex]$r$[/tex] \\
\hline
[tex]$R$[/tex] & [tex]$RR$[/tex] & [tex]$Rr$[/tex] \\
\hline
[tex]$r$[/tex] & [tex]$rR$[/tex] & [tex]$rr$[/tex] \\
\hline
\end{tabular}



Answer :

GinnyAnswer
Let's understand the inheritance pattern of pea plants for determining probabilities. We know round peas (R) are dominant over wrinkled peas (r).

### Second to Third Generation Cross
When second-generation pea plants, each with the genotype Rr, are crossed, they form the following Punnett square:

[tex]\[ \begin{array}{c|c|c} & R & r \\ \hline R & RR & Rr \\ \hline r & Rr & rr \\ \end{array} \][/tex]

This table gives us the probabilities of the various genotypes in the third generation:
- [tex]\( RR \)[/tex] (homozygous dominant - round)
- [tex]\( Rr \)[/tex] (heterozygous - round)
- [tex]\( rr \)[/tex] (homozygous recessive - wrinkled)

Probability of [tex]\(RR\)[/tex]:
[tex]\[ \text{Number of } RR \text{ genotypes} = 1 \text{ (out of 4 possibilities)} \implies P(RR) = \frac{1}{4} = 0.25 \][/tex]

Probability of [tex]\(Rr\)[/tex]:
[tex]\[ \text{Number of } Rr \text{ genotypes} = 2 \text{ (out of 4 possibilities)} \implies P(Rr) = \frac{2}{4} = 0.50 \][/tex]

Probability of round peas (either [tex]\(RR\)[/tex] or [tex]\(Rr\)[/tex]):
[tex]\[ P(\text{Round}) = P(RR) + P(Rr) = 0.25 + 0.50 = 0.75 \][/tex]

Probability of [tex]\(rr\)[/tex]:
[tex]\[ \text{Number of } rr \text{ genotypes (wrinkled)} = 1 \text{ (out of 4 possibilities)} \implies P(rr) = \frac{1}{4} = 0.25 \][/tex]

### Summary:
(a) The probability that a third-generation offspring has round peas is [tex]\(0.75\)[/tex].

(b) The probability that a third-generation offspring has wrinkled peas is [tex]\(0.25\)[/tex].

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