This problem involves the binomial probability formula since we are dealing with a sequence of independent trials (each at-bat is independent), and the outcome for each trial is either a hit or no hit.

The binomial probability formula is:

$P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$

Where:

• $n = 7$ (number of at-bats)
• $k = 2$ (number of hits)
• $p = 0.305$ (probability of a hit)
• $1 - p = 0.695$ (probability of no hit)

Now, we'll calculate this step by step.

1. The binomial coefficient $\binom{n}{k}$ is calculated as:

$\binom{7}{2} = \frac{7!}{2!(7 - 2)!} = \frac{7 \times 6}{2 \times 1} = 21$

2. Now, we calculate the probability:

$P(X = 2) = 21 \times (0.305)^2 \times (0.695)^5$

Let me calculate this for you.The probability that the baseball player gets exactly 2 hits in his next 7 at-bats is $0.3168$.

Do you have any questions or would you like more details on the steps?

Here are 5 related questions to expand on this:

1. What is the probability of the player getting exactly 3 hits in his next 7 at-bats?
2. How would the probability change if the player had a batting average of 0.400?
3. What is the probability of the player getting no hits in 7 at-bats?
4. What is the probability of getting at least 2 hits in 7 at-bats?
5. How many hits is the player expected to get in 7 at-bats?

Tip: In binomial probability, the mean number of successes can be found by multiplying the number of trials by the probability of success $\mu = n \times p$.