Sure, let's go through a detailed, step-by-step solution to understand why the genotype Tt occurs 50% of the time.
### Step-by-Step Solution:
1. Define Probabilities of Each Allele:
- Probability of allele T is [tex]\( P(T) = 0.5 \)[/tex]
- Probability of allele t is [tex]\( P(t) = 0.5 \)[/tex]
2. Determine All Possible Combinations:
When an organism has two alleles, there are four possible combinations:
- TT (both alleles are T)
- Tt (one allele is T, one allele is t)
- tT (one allele is t, one allele is T - this is essentially the same as Tt)
- tt (both alleles are t)
3. Calculate the Probability of Each Combination:
- Probability of genotype TT: [tex]\( P(TT) = P(T) \times P(T) = 0.5 \times 0.5 = 0.25 \)[/tex]
- Probability of genotype Tt: [tex]\( P(Tt) = P(T) \times P(t) = 0.5 \times 0.5 = 0.25 \)[/tex]
- Probability of genotype tT: [tex]\( P(tT) = P(t) \times P(T) = 0.5 \times 0.5 = 0.25 \)[/tex]
- Probability of genotype tt: [tex]\( P(tt) = P(t) \times P(t) = 0.5 \times 0.5 = 0.25 \)[/tex]
4. Combine the Probabilities of Heterozygous Combination (Tt and tT):
Since the heterozygous genotype Tt can occur in two ways (Tt or tT):
- [tex]\( P(Tt) + P(tT) = 0.25 + 0.25 = 0.5 \)[/tex]
5. Conclusion:
Therefore, the genotype Tt occurs 50% of the time because the combined probability of the two different ways to form Tt (Tt and tT) is 0.25 + 0.25 = 0.5, or 50%.
These probabilities add up correctly:
- TT: 0.25 (25%)
- Tt: 0.25 (25%)
- tT: 0.25 (25%)
- tt: 0.25 (25%)
The total probability is [tex]\( 0.25 + 0.25 + 0.25 + 0.25 = 1 \)[/tex] or 100% which reassures that our calculations are accurate.
Therefore, the Tt genotype occurs 50% of the time.
