Family pedigree chart illustrating genetic inheritance across three generations, with focus on affected individuals.

Problem Analysis:

We are given a family pedigree and asked to determine the mode of inheritance, the genotypes of individuals II-3 and III-1, and the probability that III-1 would have an affected son. Let's break this down step by step.

#### Step 1: Analyze the Pedigree
The pedigree shows:
- Individual I-1 (male) is affected (black square).
- Individual I-2 (female) is unaffected (white circle).
- There are three offspring in the second generation (II-1, II-2, II-3, II-4, II-5, II-6):
- II-1 and II-3 are males, unaffected.
- II-2 and II-4 are females, unaffected.
- II-5 is a female, unaffected.
- II-6 is a male, unaffected.
- In the third generation (III), there are several offspring:
- III-1 (female) is unaffected.
- III-2 (male) is affected.
- III-3 (female) is unaffected.
- III-4 (male) is unaffected.
- III-5 (female) is unaffected.
- III-6 (male) is affected.

#### Step 2: Determine the Mode of Inheritance
To determine the mode of inheritance, let's consider the key features of the pedigree:
1. Affected Individuals: Only males (III-2 and III-6) are affected.
2. Pattern of Transmission:
- The affected trait skips generations.
- Affected males do not have affected fathers.
- Unaffected females can have affected sons.
3. Key Observations:
- The trait appears to be passed through females but only manifests in males.
- This pattern is consistent with X-linked recessive inheritance.

#### Reasoning for X-Linked Recessive Inheritance:
- In X-linked recessive inheritance:
- Males have one X chromosome (XY), so if they inherit the recessive allele, they will be affected.
- Females have two X chromosomes (XX), so they need two copies of the recessive allele to be affected. If they have one copy, they are carriers but unaffected.
- The pedigree shows that:
- Affected individuals are all males.
- Unaffected females (e.g., II-2, II-4, II-5) can pass the recessive allele to their sons.
- The pattern of transmission matches X-linked recessive inheritance.

Thus, the mode of inheritance is X-linked recessive.

#### Step 3: Determine Genotypes
##### Individual II-3:
- II-3 is an unaffected male.
- In X-linked recessive inheritance, unaffected males have the genotype `X^A Y` (where `X^A` is the dominant allele and `Y` is the Y chromosome).

So, the genotype for II-3 is:
$$
\boxed{X^A Y}
$$

##### Individual III-1:
- III-1 is an unaffected female.
- Since her brother (III-2) is affected (`X^a Y`), her mother (II-2) must be a carrier (`X^A X^a`).
- III-1 could have inherited either `X^A` or `X^a` from her mother. However, since she is unaffected, her genotype must be `X^A X^a`.

So, the genotype for III-1 is:
$$
\boxed{X^A X^a}
$$

#### Step 4: Probability of III-1 Having an Affected Son
- III-1 has the genotype `X^A X^a`.
- Her husband (not shown in the pedigree) is assumed to be unaffected, so his genotype is `X^A Y`.
- The possible offspring genotypes from `X^A X^a` (III-1) and `X^A Y` (husband) are:
- Sons: `X^A Y` (unaffected), `X^a Y` (affected)
- Daughters: `X^A X^A` (unaffected), `X^A X^a` (carrier)

The probability of having an affected son (`X^a Y`) is:
$$
\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
$$

However, since we are only considering sons, the probability simplifies to:
$$
\frac{1}{2}
$$

So, the probability that III-1 would have an affected son is:
$$
\boxed{\frac{1}{2}}
$$

#### Final Answer Summary:
1. Mode of inheritance: X-linked recessive
2. Genotype for II-3: \( X^A Y \)
3. Genotype for III-1: \( X^A X^a \)
4. Probability that III-1 would have an affected son: \( \frac{1}{2} \)

$$
\boxed{X\text{-linked recessive, } X^A Y, X^A X^a, \frac{1}{2}}
$$