The famous villain, Gabbar Singh, from the film ‘Sholey’ has now grown pretty old. So the probability that he can shoot a target at 10m distance is 0.75 only. 6 of his subordinate goons have been beaten by Jai and Veeru and are now standing before Gabbar. As a punishment, he asks each of them to stand on a stone pedestal 10m away from him and takes a shot at them. There are 6 bullets in the revolver and these 6 people will all be shot at exactly once one after another. If the bullet hits anyone (with probability 0.75), that person dies, else he is forgiven. The probability that after all rounds 2 will be dead is p = x/(2^12) then find x

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 \( \boxed{135} \).

---

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.

The famous villain, Gabbar Singh, from the film ‘Sholey’ has now grown pretty old. So the probability that he can shoot a target at 10m distance is 0.75 only. 6 of his subordinate goons have been beaten by Jai and Veeru and are now standing before Gabbar. As a punishment, he asks each of them to stand on a stone pedestal 10m away from him and takes a shot at them. There are 6 bullets in the revolver and these 6 people will all be shot at exactly once one after another. If the bullet hits anyone (with probability 0.75), that person dies, else he is forgiven. The probability that after all rounds 2 will be dead is p = x/(2^12) then find x.

Solve a probability problem involving Gabbar Singh's shooting accuracy using binomial distribution. Learn to calculate the probability of exactly 2 hits out of 6 shots with a 0.75 chance of success, using key binomial formulas and concepts. Suitable for Grades 10-12.

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