91Ó°ÊÓ

Each question has 4 possible choices, and only one is correct. Therefore, the probability that Robin guesses an answer correctly is \( \frac{1}{4} \). The probability of guessing incorrectly is \( \frac{3}{4} \).

For Robin to get the first correct answer on the 3rd question, she must get the first two questions wrong, followed by a correct answer on the 3rd question.This is calculated using: \( P = \left( \frac{3}{4} \right)^2 \times \frac{1}{4} \). Calculating gives: \[ P = \left( \frac{3}{4} \right)^2 \times \frac{1}{4} = \frac{9}{64} \].

To find the probability that Robin gets exactly 3 or 4 questions right, we use the binomial probability formula: \( P(X = k) = \binom{n}{k} (p^k) (1-p)^{n-k} \), where \( n = 5 \), \( p = \frac{1}{4} \), and \( k \) is the number of correct answers.1. For exactly 3 correct: \( P(X = 3) = \binom{5}{3} \left(\frac{1}{4}\right)^3 \left(\frac{3}{4}\right)^2 \). - Calculate: \( \binom{5}{3} = 10 \), so \( P = 10 \cdot \left(\frac{1}{4}\right)^3 \cdot \left(\frac{3}{4}\right)^2 = \frac{135}{1024} \).2. For exactly 4 correct: \( P(X = 4) = \binom{5}{4} \left(\frac{1}{4}\right)^4 \left(\frac{3}{4}\right)^1 \). - Calculate: \( \binom{5}{4} = 5 \), so \( P = 5 \cdot \left(\frac{1}{4}\right)^4 \cdot \frac{3}{4} = \frac{15}{1024} \).3. Total probability for exactly 3 or 4 correct answers: \( P = \frac{135}{1024} + \frac{15}{1024} = \frac{150}{1024} = \frac{75}{512} \).

To get the majority of right answers, Robin must get at least 3 correct answers.Using the previous calculations:- Probability of exactly 3 correct answers is \( \frac{135}{1024} \).- Probability of exactly 4 correct answers is \( \frac{15}{1024} \).- Probability of all 5 correct is \( P(X = 5) = \binom{5}{5} \left(\frac{1}{4}\right)^5 = \frac{1}{1024} \).Sum all probabilities: \( P = \frac{135}{1024} + \frac{15}{1024} + \frac{1}{1024} = \frac{151}{1024} \).