Define the total number of marbles and the probability of picking two black marbles.

There are 7 black marbles and $x$ red marbles.
The total number of marbles is $7+x$.
The probability of picking two black marbles without replacement is the product of the probability of picking the first black marble and the probability of picking the second black marble (given the first was black):

We are given that the probability of both marbles being black is $\frac{7}{22}$.

Divide both sides by 7:

Cross-multiply:

Rearrange into a quadratic equation:

We look for two numbers that multiply to -90 and add to 13. These numbers are 18 and -5.

This gives two possible values for $x$: $x = -18$ or $x = 5$. Since the number of marbles cannot be negative, we choose the positive value.
The value of $x$ is $\boxed{\text{5}}$.

2. The probability of picking one marble of each color.

With $x=5$:
Number of black marbles = 7.
Number of red marbles = 5.
Total number of marbles = $7+5 = 12$.

There are two mutually exclusive ways to pick one marble of each color:

• Picking a black marble first, then a red marble ($P(\text{B then R})$).
• Picking a red marble first, then a black marble ($P(\text{R then B})$).

The total probability of picking one marble of each color is the sum of these two probabilities:

Simplify the fraction by dividing the numerator and denominator by 2:

The probability of picking one marble of each color is $\boxed{\frac{35}{66}}$.