Raj needs to score at least 70% on the test to pass, which means he needs to answer at least 7 out of the 10 questions correctly. Since he is guessing randomly and each question has 5 possible choices, the probability of guessing a correct answer is:

$p = \frac{1}{5} = 0.2$

Similarly, the probability of guessing incorrectly is:

$q = 1 - p = 0.8$

This problem can be modeled using the binomial distribution, where the probability of getting exactly $k$ correct answers in 10 trials (questions) is given by:

$P(X = k) = \binom{10}{k} p^k q^{10-k}$

Where:

• $\binom{10}{k}$ is the binomial coefficient, representing the number of ways to choose $k$ correct answers from 10,
• $p = 0.2$ is the probability of getting a correct answer,
• $q = 0.8$ is the probability of getting an incorrect answer.

Raj needs to get at least 7 correct answers. Therefore, we need to calculate the cumulative probability of getting 7, 8, 9, or 10 correct answers:

$P(\text{pass}) = P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)$

Let me calculate this probability for you.The probability that Raj will pass the test by guessing at random and getting at least 7 out of 10 questions correct is approximately 0.000864, or 0.0864%.

Would you like further clarification on this, or any additional details?

Here are 5 related questions you might find interesting:

1. How does the probability change if Raj needs to score 60% instead of 70%?
2. What if Raj had some knowledge and his probability of guessing correctly increased to 0.4?
3. How does the number of choices per question affect the probability of passing?
4. What is the expected number of correct answers Raj will get if he guesses randomly?
5. How would the probability change if there were 15 questions instead of 10?

Tip: Binomial distribution is useful for modeling the number of successes in a fixed number of independent trials, like guessing answers on a multiple-choice test.