To solve this, we need to use the binomial probability distribution since the number of hits in the game follows a binomial pattern: each at-bat is an independent trial with two outcomes (hit or no hit). Here's how we proceed:

Problem Breakdown:

• The player's batting average (probability of getting a hit in a given at-bat) is $p = 0.182$.
• The number of at-bats (trials) in the game is $n = 5$.
• We need to find the probability that the player gets at least 2 hits in 5 at-bats, which means the player gets 2 or more hits.

Let $X$ be the random variable representing the number of hits the player gets. We are looking for:

$P(X \geq 2)$

This is equivalent to:

$P(X \geq 2) = 1 - P(X < 2) = 1 - (P(X = 0) + P(X = 1))$

We'll now calculate $P(X = 0)$ and $P(X = 1)$ using the binomial probability formula:

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

Where $\binom{n}{k}$ is the binomial coefficient and represents the number of ways to choose $k$ successes (hits) from $n$ trials (at-bats).

Step 1: Compute $P(X = 0)$

$P(X = 0) = \binom{5}{0} (0.182)^0 (1 - 0.182)^5 = 1 \times 1 \times (0.818)^5$

$P(X = 0) = (0.818)^5 \approx 0.340$

Step 2: Compute $P(X = 1)$

$P(X = 1) = \binom{5}{1} (0.182)^1 (1 - 0.182)^4 = 5 \times 0.182 \times (0.818)^4$

$P(X = 1) = 5 \times 0.182 \times 0.379 \approx 0.345$

Step 3: Compute $P(X \geq 2)$

Now, we can calculate the probability of getting at least 2 hits:

$P(X \geq 2) = 1 - (P(X = 0) + P(X = 1)) = 1 - (0.340 + 0.345) = 1 - 0.685 = 0.315$

Final Answer:

The probability that the player will get at least 2 hits in the game is approximately 0.315 or 31.5%.

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Would you like more details on any part of the calculation, or do you have other questions? Here are some related topics you might find interesting:

1. What is the binomial probability distribution and when do we use it?
2. How do we calculate binomial coefficients?
3. What are the key properties of a binomial distribution?
4. How does changing the probability of success affect the outcome of a binomial distribution?
5. Can we use this method to calculate the probability of exactly 3 hits or more?

Tip: When working with binomial probabilities, always remember to check if the trials are independent and if the probability of success remains constant for each trial.