Let's solve this problem step by step.

Problem Breakdown:

• Gabbar's probability of hitting a target is $P(\text{hit}) = 0.75$.
• The probability of missing a target is $P(\text{miss}) = 1 - 0.75 = 0.25$.
• There are 6 people, and Gabbar shoots once at each.
• We are asked to find the probability that exactly 2 of them are hit and die.

This is a binomial probability problem because we have a fixed number of trials (6 shots), each with two possible outcomes: success (hit) or failure (miss).

The binomial probability formula is:

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

where:

• $n$ is the number of trials (6 in this case),
• $k$ is the number of successful hits (2 in this case),
• $p$ is the probability of success (0.75).

Step 1: Calculate the binomial coefficient

We need to calculate the binomial coefficient $\binom{6}{2}$, which is the number of ways to choose 2 successes (hits) out of 6 trials (shots). This is given by:

$\binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15$

Step 2: Calculate the probability of 2 hits

The probability of exactly 2 hits is:

$P(X = 2) = \binom{6}{2} \times (0.75)^2 \times (0.25)^4$

Substitute the values:

$P(X = 2) = 15 \times (0.75)^2 \times (0.25)^4$

Now, calculate the powers of 0.75 and 0.25:

$0.75^2 = 0.5625 \quad \text{and} \quad 0.25^4 = 0.00390625$

Substitute these values back:

$P(X = 2) = 15 \times 0.5625 \times 0.00390625$

Step 3: Perform the multiplication

First, multiply $15 \times 0.5625$:

$15 \times 0.5625 = 8.4375$

Next, multiply by $0.00390625$:

$P(X = 2) = 8.4375 \times 0.00390625 = 0.0329437255859375$

Step 4: Express in the required form

We are given that the probability is expressed as $\frac{x}{2^{12}}$. Since $2^{12} = 4096$, we write:

$P(X = 2) = \frac{x}{4096}$

Thus:

$\frac{x}{4096} = 0.0329437255859375$

Multiply both sides by 4096:

$x = 0.0329437255859375 \times 4096 = 134.94$

Rounding to the nearest integer, we get:

$x = 135$

Final Answer:

The value of $x$ is $\boxed{135}$.

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Would you like more details or have any questions?

Here are 5 related questions you can try:

1. What is the probability that all 6 people survive?
2. How would the probability change if Gabbar's accuracy was 0.8 instead of 0.75?
3. What is the probability that exactly 3 people are hit?
4. What is the expected number of people Gabbar will hit?
5. How does the binomial distribution change as the number of people increases?

Tip: When working with binomial distributions, it helps to remember that the sum of probabilities for all outcomes must equal 1. This can be useful for checking your work.