11. A test paper consists of 40 questions. Each question is followed by four choices of answer, where only one of these is correct. (a) Salma answers all the questions by randomly choosing an answer for each question. $$ \begin{array}{ll} ext { (i) Estimate the number of questions she answered correctly. } 10 \ ext { (ii) Find the standard deviation of the number of questions } \ ext { she answered correctly. } & 2-739 \ ext { [b) Basri answers } 30 ext { questions correctly and randemly choose an } \ ext { answer for each of the remaining } 10 ext { questions. Find the } \ ext { probability that he answers } & ext { (i) } 36 ext { questions correctly, } \ ext { (ii) at least } 32 ext { questions cornectly. } & 0.01622 ext { is marks] } \ 6.756 & \end{array} $$

Question

11. A test paper consists of 40 questions. Each question is followed by four choices of answer, where only one of these is correct. (a) Salma answers all the questions by randomly choosing an answer for each question. $$ \begin{array}{ll}	ext { (i) Estimate the number of questions she answered correctly. } 10 \ 	ext { (ii) Find the standard deviation of the number of questions } \ 	ext { she answered correctly. } & 2-739 \ 	ext { [b) Basri answers } 30 	ext { questions correctly and randemly choose an } \ 	ext { answer for each of the remaining } 10 	ext { questions. Find the } \ 	ext { probability that he answers } & 	ext { (i) } 36 	ext { questions correctly, } \ 	ext { (ii) at least } 32 	ext { questions cornectly. } & 0.01622 	ext { is marks] } \ 6.756 & \end{array} $$

UpStudy AI · Accepted Answer

**(a) Salma’s Answers by Random Guessing**

1. **Expected Number of Correct Answers**  
   Each question has a probability of success  
   $$
   p = \frac{1}{4}
   $$  
   Since there are 40 independent questions, the expected number is  
   $$
   E(X) = 40 \times \frac{1}{4} = 10.
   $$

2. **Standard Deviation**  
   For a binomial distribution with parameters $$ n = 40 $$ and $$ p = \frac{1}{4} $$, the variance is  
   $$
   \text{Var}(X) = n \cdot p \cdot (1-p) = 40 \times \frac{1}{4} \times \frac{3}{4} = 40 \times \frac{3}{16} = \frac{120}{16} = 7.5.
   $$  
   Thus, the standard deviation is  
   $$
   \sigma = \sqrt{7.5} \approx 2.74.
   $$

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**(b) Basri’s Answers with 30 Fixed Correct Answers**

Basri answers 30 questions correctly for sure. For the remaining 10 questions, he guesses. Let $$ X $$ be the number of additional correct answers from these 10 questions. Here, $$ X $$ follows a binomial distribution with parameters $$ n = 10 $$ and $$ p = \frac{1}{4} $$.

1. **(i) Probability of Exactly 36 Correct Answers**

To have a total of 36 correct answers, Basri must score exactly 6 correct answers from the 10 guessed questions (since $$ 30 + 6 = 36 $$). Therefore, the probability is given by  
   $$
   P(X = 6) = \binom{10}{6}\left(\frac{1}{4}\right)^6\left(\frac{3}{4}\right)^4.
   $$

2. **(ii) Probability of At Least 32 Correct Answers**

To obtain at least 32 correct answers in total, Basri must get at least 2 correct answers from the 10 guessed questions (since $$ 30 + 2 = 32 $$). Thus, we need  
   $$
   P(X \geq 2) = 1 - \left[P(X=0) + P(X=1)\right].
   $$
   
   Where:  
   $$
   P(X=0) = \binom{10}{0}\left(\frac{1}{4}\right)^0\left(\frac{3}{4}\right)^{10} = \left(\frac{3}{4}\right)^{10},
   $$
   $$
   P(X=1) = \binom{10}{1}\left(\frac{1}{4}\right)^1\left(\frac{3}{4}\right)^9 = 10 \cdot \frac{1}{4} \cdot \left(\frac{3}{4}\right)^9.
   $$
   
   Therefore,  
   $$
   P(X \geq 2) = 1 - \left[\left(\frac{3}{4}\right)^{10} + 10 \cdot \frac{1}{4} \cdot \left(\frac{3}{4}\right)^9\right].
   $$

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**Final Answers**

- **(a-i)** $$\displaystyle E(X) = 10$$  
- **(a-ii)** $$\displaystyle \sigma \approx 2.74$$  
- **(b-i)** $$\displaystyle P(36\text{ correct}) = \binom{10}{6}\left(\frac{1}{4}\right)^6\left(\frac{3}{4}\right)^4$$  
- **(b-ii)** $$\displaystyle P(\text{at least } 32\text{ correct}) = 1 - \left[\left(\frac{3}{4}\right)^{10} + 10 \cdot \frac{1}{4} \cdot \left(\frac{3}{4}\right)^9\right]$$
**(a) Salma’s Answers by Random Guessing** - **(i) Expected Number of Correct Answers:** 10 - **(ii) Standard Deviation:** Approximately 2.74 **(b) Basri’s Answers with 30 Fixed Correct Answers** - **(i) Probability of Exactly 36 Correct Answers:** Calculated using the binomial probability formula. - **(ii) Probability of At Least 32 Correct Answers:** Calculated as $$1 - \left[\left(\frac{3}{4}\right)^{10} + 10 \cdot \frac{1}{4} \cdot \left(\frac{3}{4}\right)^9\right]$$

A test paper consists of 40 questions. Each question is followed
by four choices of answer, where only one of these is correct.
(a) Salma answers all the questions by randomly choosing an
answer for each question.

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