To answer the questions, let’s first summarize the data by the total number of individuals in each category:
- Gene 1 Dominant, Gene 2 Dominant (DD): 57
- Gene 1 Dominant, Gene 2 Recessive (DR): 24
- Gene 1 Recessive, Gene 2 Dominant (RD): 15
- Gene 1 Recessive, Gene 2 Recessive (RR): 4
Total individuals: 57 + 24 + 15 + 4 = 100
(a) What is the probability that in a randomly sampled individual, gene 1 is recessive?
The number of individuals where gene 1 is recessive consists of the individuals categorized as "Recessive (RD) and Recessive (RR)":
\[ \text{Number of individuals with gene 1 recessive} = 15 + 4 = 19 \]
The probability \( P(\text{Gene 1 is recessive}) \) is: \[ P(\text{Gene 1 is recessive}) = \frac{\text{Number of individuals with gene 1 recessive}}{\text{Total number of individuals}} = \frac{19}{100} = 0.19 \]
(b) What is the probability that in a randomly sampled individual, gene 2 is recessive?
The number of individuals where gene 2 is recessive consists of the individuals categorized as "Recessive (DR) and Recessive (RR)":
\[ \text{Number of individuals with gene 2 recessive} = 24 + 4 = 28 \]
The probability \( P(\text{Gene 2 is recessive}) \) is: \[ P(\text{Gene 2 is recessive}) = \frac{\text{Number of individuals with gene 2 recessive}}{\text{Total number of individuals}} = \frac{28}{100} = 0.28 \]
(c) Given that gene 1 is recessive, what is the probability that gene 2 is recessive?
To find this conditional probability \( P(\text{Gene 2 is recessive} | \text{Gene 1 is recessive}) \), we use:
\[ P(\text{Gene 2 is recessive} | \text{Gene 1 is recessive}) = \frac{P(\text{Gene 1 is recessive and Gene 2 is recessive})}{P(\text{Gene 1 is recessive})} \]
From part (a), we have \( P(\text{Gene 1 is recessive}) = 0.19 \).
The number of individuals with both gene 1 and gene 2 recessive is 4 (from the RR category).
Thus, the probability \( P(\text{Gene 1 is recessive and Gene 2 is recessive}) = \frac{4}{100} = 0.04 \).
Now substituting these values:
\[ P(\text{Gene 2 is recessive} | \text{Gene 1 is recessive}) = \frac{0.04}{0.19} \approx 0.2105 \]
(d) Are these genes in linkage equilibrium?
To check if gene 1 recessive is independent of gene 2 recessive, we need to see if the following holds:
\[ P(\text{Gene 2 is recessive} | \text{Gene 1 is recessive}) = P(\text{Gene 2 is recessive}) \]
From above, we have:
- \( P(\text{Gene 2 is recessive} | \text{Gene 1 is recessive}) \approx 0.2105 \)
- \( P(\text{Gene 2 is recessive}) = 0.28 \)
Since \( 0.2105 \neq 0.28 \), we conclude that the events are not independent.
Thus, the genes are not in linkage equilibrium.
Summary:
(a) Probability that gene 1 is recessive: 0.19
(b) Probability that gene 2 is recessive: 0.28
(c) Probability that gene 2 is recessive given gene 1 is recessive: 0.2105
(d) The genes are not in linkage equilibrium.