There are 5 questions in a multiple choice examination in which each question has 3 possible answers.Find the probability that a student gives 4 correct answers by guess only.

Explanation

To solve the problem of finding the probability that a student gives 4 correct answers by guessing in a multiple-choice examination with 5 questions, each having 3 possible answers, we can follow these steps:

### Step 1: Identify the total number of questions and possible answers
We have:
- Total number of questions (n) = 5
- Number of possible answers for each question = 3

### Step 2: Determine the probability of getting a correct answer
The probability of guessing a correct answer (p) is:
\[ p = \frac{1}{3} \]
Since there are 3 options and only one is correct.

### Step 3: Determine the probability of getting an incorrect answer
The probability of guessing an incorrect answer (q) is:
\[ q = 1 - p = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 4: Use the binomial probability formula
We want to find the probability of getting exactly 4 correct answers out of 5 questions. This scenario can be modeled using the binomial probability formula:
\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]
where:
- \( n \) = total number of trials (questions) = 5
- \( k \) = number of successful trials (correct answers) = 4
- \( p \) = probability of success (correct answer) = \( \frac{1}{3} \)
- \( q \) = probability of failure (incorrect answer) = \( \frac{2}{3} \)

### Step 5: Calculate the binomial coefficient
Calculate \( \binom{5}{4} \):
\[ \binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5!}{4! \cdot 1!} = 5 \]

### Step 6: Substitute values into the formula
Now, substitute the values into the binomial probability formula:
\[ P(X = 4) = \binom{5}{4} p^4 q^{5-4} \]
\[ P(X = 4) = 5 \left(\frac{1}{3}\right)^4 \left(\frac{2}{3}\right)^1 \]

### Step 7: Calculate \( p^4 \) and \( q^1 \)
Calculate \( \left(\frac{1}{3}\right)^4 \):
\[ \left(\frac{1}{3}\right)^4 = \frac{1}{81} \]
Calculate \( \left(\frac{2}{3}\right)^1 \):
\[ \left(\frac{2}{3}\right)^1 = \frac{2}{3} \]

### Step 8: Combine the results
Now plug these values back into the equation:
\[ P(X = 4) = 5 \cdot \frac{1}{81} \cdot \frac{2}{3} \]
\[ P(X = 4) = 5 \cdot \frac{2}{243} \]
\[ P(X = 4) = \frac{10}{243} \]

### Final Answer
The probability that a student gives 4 correct answers by guessing is:
\[ \frac{10}{243} \]