The task involves calculating probabilities in a binomial distribution setting. Here's a detailed breakdown of the problem:

Given:

• Probability of hitting the target in one shot: $p = 0.9$.
• Number of shots: $n = 7$.
• Complement probability (miss): $q = 1 - p = 0.1$.

You are asked to calculate the following:

1. The probability of hitting the target at least 2 times in 7 shots.
2. The probability of hitting the target at least 1 time in 7 shots.

Solution:

The binomial probability formula is: $P(X = k) = C(n, k) \cdot p^k \cdot q^{n-k},$ where:

• $C(n, k) = \frac{n!}{k!(n-k)!}$ is the number of combinations,
• $p$ is the probability of success,
• $q = 1-p$ is the probability of failure,
• $k$ is the number of successes.

To compute "at least" probabilities, we use: $P(X \geq k) = 1 - P(X < k) = 1 - \sum_{i=0}^{k-1} P(X = i).$

1) Probability of hitting at least 2 times:

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

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

$P(X = 0) = C(7, 0) \cdot (0.9)^0 \cdot (0.1)^7 = 1 \cdot 1 \cdot 0.0000001 = 0.0000001.$

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

$P(X = 1) = C(7, 1) \cdot (0.9)^1 \cdot (0.1)^6 = 7 \cdot 0.9 \cdot 0.000001 = 0.0000063.$

Step 3: Subtract:

$P(X \geq 2) = 1 - P(X = 0) - P(X = 1) = 1 - 0.0000001 - 0.0000063 \approx 0.9999936.$

2) Probability of hitting at least 1 time:

$P(X \geq 1) = 1 - P(X = 0).$

Step 1: Use $P(X = 0)$ from above:

$P(X \geq 1) = 1 - 0.0000001 = 0.9999999.$

Final Answers:

1. $P(X \geq 2) \approx 0.9999936$,
2. $P(X \geq 1) \approx 0.9999999$.

Would you like me to provide step-by-step calculations or visualizations for binomial probabilities?

Related Questions:

1. What is the probability of hitting the target exactly 3 times in 7 shots?
2. How do you calculate $P(X = k)$ for any given $k$ in a binomial distribution?
3. What are the expected value and variance of this binomial experiment?
4. How do probabilities change if $p$ is reduced to 0.8 instead of 0.9?
5. How does the complement rule simplify probability calculations?

Tip:

When solving binomial problems, knowing the cumulative probabilities (using tables or software) can save significant time for larger values of $n$.