To determine the probability of an offspring being tall and having purple flowers when two plants heterozygous for both traits (TtPp) are crossed, we need to use a Punnett square to analyze the genetic combinations. The two traits (plant height and flower color) are segregated independently:
- Tall (T) is dominant over short (t).
- Purple (P) is dominant over white (p).
When crossing two TtPp plants, each parent can produce four types of gametes: TP, Tp, tP, and tp. Therefore, the Punnett square for this cross has 16 possible outcomes.
Below is a Punnett square for the cross TtPp x TtPp:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline & TP & Tp & tP & tp \\
\hline
TP & TTPP & TTPp & TtPP & TtPp \\
\hline
Tp & TTPp & TTpp & TtPp & Ttpp \\
\hline
tP & TtPP & TtPp & ttPP & ttPp \\
\hline
tp & TtPp & Ttpp & ttPp & ttpp \\
\hline
\end{array}
\][/tex]
To find out how many offspring will be tall (having at least one T allele) and have purple flowers (having at least one P allele), we look for the combinations in the grid that include at least one T and one P:
[tex]\[
\begin{aligned}
\text{TTPP} & \quad (\text{Tall, Purple})\\
\text{TTPp} & \quad (\text{Tall, Purple})\\
\text{TtPP} & \quad (\text{Tall, Purple})\\
\text{TtPp} & \quad (\text{Tall, Purple})\\
\end{aligned}
\][/tex]
So, the combinations in the Punnett square result in the following tall and purple offspring:
1. TTPP
2. TTPP
3. TTPp
4. TTPp
5. TtPP
6. TtPP
7. TtPp
8. TtPp
9. TtPp
In total, there are 9 combinations out of the 16 possible combinations that will result in an offspring being tall and having purple flowers.
Thus, the probability [tex]\( P \)[/tex] of an offspring being tall and having purple flowers is calculated as follows:
[tex]\[
P = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{9}{16} = 0.5625
\][/tex]
Therefore, the probability of an offspring being tall and having purple flowers is [tex]\( \boxed{0.5625} \)[/tex].
