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Let's solve the problems step by step.

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### **5.2: For a binomial distribution, mean is 2, and variance is 1. Estimate \( n \), \( p \), and \( q \).**
We know the following for a binomial distribution:
- Mean \( \mu = np = 2 \)
- Variance \( \sigma^2 = npq = 1 \)
- \( p + q = 1 \)

From \( p + q = 1 \), \( q = 1 - p \). Substitute \( q \) into the variance equation:

\[
npq = np(1 - p) = 1
\]

Substitute \( np = 2 \) into the equation:

\[
2(1 - p) = 1
\]

Simplify:

\[
2 - 2p = 1 \implies 2p = 1 \implies p = 0.5
\]

Now, \( q = 1 - p = 1 - 0.5 = 0.5 \).

Finally, find \( n \) using \( np = 2 \):

\[
n \cdot 0.5 = 2 \implies n = 4
\]

**Solution:**
- \( n = 4 \)
- \( p = 0.5 \)
- \( q = 0.5 \)

---

### **5.3: 80% electric circuits in AIUB work properly. One day, 10 circuits were selected randomly. What was the expected number of devices working properly and its variance?**

For a binomial distribution:
- Probability of success \( p = 0.8 \),
- Number of trials \( n = 10 \).

**Expected value (mean):**

\[
\mu = np = 10 \cdot 0.8 = 8
\]

**Variance:**

\[
\sigma^2 = npq = 10 \cdot 0.8 \cdot (1 - 0.8) = 10 \cdot 0.8 \cdot 0.2 = 1.6
\]

**Solution:**
- Expected number of devices working properly: \( 8 \)
- Variance: \( 1.6 \)

---

### **5.5: 70% of all business startups in the IT industry report that they generate a profit in their first year. If a sample of 10 new IT business startups is selected, find the probability that exactly 7 will generate a profit.**

This is a binomial probability problem with:
- \( n = 10 \),
- \( p = 0.7 \),
- \( q = 1 - p = 0.3 \).

The probability of exactly \( k = 7 \) successes is given by:

\[
P(X = k) = \binom{n}{k} p^k q^{n-k}
\]

Substitute the values:

\[
P(X = 7) = \binom{10}{7} (0.7)^7 (0.3)^3
\]

Calculate step by step:
- \( \binom{10}{7} = \frac{10!}{7!(10-7)!} = \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120 \),
- \( (0.7)^7 = 0.0823543 \),
- \( (0.3)^3 = 0.027 \).

Now:

\[
P(X = 7) = 120 \cdot 0.0823543 \cdot 0.027 \approx 0.2668
\]

**Solution:**
The probability that exactly 7 startups will generate a profit is approximately \( 0.267 \) (26.7%).

---

Would you like me to explain any step in detail? Here are related questions to further explore:

1. What is the general formula for binomial mean and variance?
2. How does the probability distribution change if \( p \) decreases in Problem 5.2?
3. Can we calculate cumulative probabilities for Problem 5.5?
4. How does variance interpret the spread of data in Problem 5.3?
5. How would the results change in Problem 5.2 if the variance was doubled?

**Tip:** For binomial problems, always identify \( n \), \( p \), and \( q \) first before applying the formulas.

Step-by-step solutions for binomial distribution problems, including estimating parameters n, p, and q, calculating expected values and variances, and finding specific probabilities. Ideal for high school students learning advanced probability concepts.

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