159. When coin is rolled three times find the probability that exactly one head will appear?
a) 0.375 b) 0.625 c) 0.357 d) 0.875
160. One empire from Sindh, two from Baluchistan, two from Punjab and four from KPK, what is the probability 
that one is from each province?
a) 6.8% b) 12.7% c) 9.8% d) None of these
161. The average performance of the students of a college over last 10 years shows that the percentages of 
students securing A, B and C grades are 10%, 40% and 50% respectively. The current year’s result shows that out 
of total 400 students, the number of students securing A, B and C grades aggregated to 64, 144 and 192 
respectively. Calculated value of chi square for this data would be?
a) 16.32 b) 2.24 c) 8.63 d) 8.16
162. A sample of 100 employees of an entity had a mean weight of 170 pounds with a standard deviation of 5 
pound. Which of the following sample sizes will result in a confidence interval of the mean ±3 pound for this data 
at the 99% confidence interval?
a) 3 b) 280 c) 11 d) 19

Let's go through each of the four questions step by step.

---

### 159. Probability of getting exactly one head when rolling a coin three times.

There are 3 trials (coin rolls) and the probability of getting heads (H) or tails (T) is equal, i.e., \( P(H) = P(T) = 0.5 \).

- **Sample Space**: When rolling a coin 3 times, there are \( 2^3 = 8 \) possible outcomes:  
  \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}

- **Event of exactly 1 head**: To get exactly one head, the possible outcomes are:  
  \{HTT, THT, TTH\}.

Thus, the number of favorable outcomes is 3. So, the probability is:

\[
P(\text{exactly 1 head}) = \frac{3}{8} = 0.375
\]

**Answer**: \( \boxed{a) 0.375} \)

---

### 160. Probability that one empire is from each province.

We are given:
- 1 empire from Sindh
- 2 empires from Baluchistan
- 2 empires from Punjab
- 4 empires from KPK

We are tasked to find the probability that one empire is selected from each of the provinces.

- **Total combinations**: We are selecting 1 empire from each province, so the total number of ways to make this selection is:
  \[
  1 \times 2 \times 2 \times 4 = 16
  \]

- **Total possible combinations**: Since there are 9 empires in total, the total possible selections is:
  \[
  \binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126
  \]

- **Probability**: The probability of selecting 1 empire from each province is:
  \[
  P = \frac{16}{126} = \frac{8}{63} \approx 0.127
  \]

**Answer**: \( \boxed{b) 12.7\%} \)

---

### 161. Chi-square calculation based on student grades.

The observed data for 400 students is:
- A: 64 students
- B: 144 students
- C: 192 students

The expected percentages are:
- A: 10% of 400 = 40
- B: 40% of 400 = 160
- C: 50% of 400 = 200

Now, we use the chi-square formula:
\[
\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}
\]
where \( O_i \) is the observed frequency and \( E_i \) is the expected frequency.

For grade A:
\[
\frac{(64 - 40)^2}{40} = \frac{24^2}{40} = \frac{576}{40} = 14.4
\]

For grade B:
\[
\frac{(144 - 160)^2}{160} = \frac{16^2}{160} = \frac{256}{160} = 1.6
\]

For grade C:
\[
\frac{(192 - 200)^2}{200} = \frac{8^2}{200} = \frac{64}{200} = 0.32
\]

Summing these values:
\[
\chi^2 = 14.4 + 1.6 + 0.32 = 16.32
\]

**Answer**: \( \boxed{a) 16.32} \)

---

### 162. Sample size for a confidence interval of ±3 pounds at 99% confidence level.

We know:
- Mean = 170 pounds
- Standard deviation (σ) = 5 pounds
- Desired margin of error (E) = ±3 pounds
- Confidence level = 99%

The critical value for a 99% confidence interval (z-score) is approximately 2.576. We use the formula for the margin of error in a confidence interval:
\[
E = z \times \frac{\sigma}{\sqrt{n}}
\]

Substituting the values:
\[
3 = 2.576 \times \frac{5}{\sqrt{n}}
\]
\[
\sqrt{n} = \frac{2.576 \times 5}{3} \approx 4.293
\]
\[
n \approx (4.293)^2 \approx 18.43
\]

So, the closest integer is \( n = 19 \).

**Answer**: \( \boxed{d) 19} \)

---

### Do you have any questions or need more details on any of these solutions?

### Related Questions:
1. What is the probability of getting exactly two heads in three coin tosses?
2. How would the probability change if 2 empires were selected from each province?
3. How is chi-square used to measure the goodness of fit?
4. How does the margin of error change if the confidence level is 95%?
5. What is the significance of the z-score in confidence intervals?

### Tip:
For probability questions involving multiple trials, it's helpful to think of them in terms of binomial distributions to get probabilities for "exactly k successes."

159. When coin is rolled three times find the probability that exactly one head will appear? a) 0.375 b) 0.625 c) 0.357 d) 0.875 160. One empire from Sindh, two from Baluchistan, two from Punjab and four from KPK, what is the probability that one is from each province? a) 6.8% b) 12.7% c) 9.8% d) None of these 161. The average performance of the students of a college over last 10 years shows that the percentages of students securing A, B and C grades are 10%, 40% and 50% respectively. The current year’s result shows that out of total 400 students, the number of students securing A, B and C grades aggregated to 64, 144 and 192 respectively. Calculated value of chi square for this data would be? a) 16.32 b) 2.24 c) 8.63 d) 8.16 162. A sample of 100 employees of an entity had a mean weight of 170 pounds with a standard deviation of 5 pound. Which of the following sample sizes will result in a confidence interval of the mean ±3 pound for this data at the 99% confidence interval? a) 3 b) 280 c) 11 d) 19

Explore how to calculate the probability of getting exactly one head in three coin tosses, the chi-square test for student performance, and the confidence interval for a sample mean. Detailed solutions cover core concepts in probability, combinatorics, and inferential statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation