91Ó°ÊÓ

In this exercise, we deal with binomial probability. Binomial probability describes the likelihood of obtaining a fixed number of successful outcomes in a set number of trials. Each trial must have exactly two possible outcomes: success or failure. This is exactly what you encounter with multiple-choice questions, where you either get the question correct or incorrect. You can calculate binomial probabilities using the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
\binom{n}{k}\ is the binomial coefficient, which represents the number of ways to choose k successes out of n trials. This formula helps find the probability of getting exactly k successes (correct answers in our case) out of n trials.

A random variable is a numerical outcome of a random phenomenon. In our problem, we define the random variable X to represent the number of correct answers out of 8 multiple-choice questions. Since we are dealing with binomial experiments, our random variable X can take integer values from 0 to 8. We denote that X follows a binomial distribution as X ~ B(8, 0.20), where 8 is the number of trials and 0.20 is the probability of success for each trial.

The binomial coefficient, noted as \binom{n}{k}\, represents the number of ways to choose k successes from n trials without regard to order. It is calculated using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
For example, to find \binom{8}{2}\, we calculate \frac{8!}{2!(8-2)!}\ = 28. This means there are 28 different ways to get exactly 2 correct answers out of 8 questions. Understanding how to calculate this coefficient is crucial for applying the binomial probability formula.

To solve the problem of finding the probability of getting fewer than 3 correct answers, we follow a step-by-step approach. First, we calculate the individual probabilities for getting 0, 1, and 2 correct answers.
\textbf{Probability for 0 Correct Answers:}\ \[ P(X = 0) = \binom{8}{0} (0.20)^0 (0.80)^8 = 0.1678 \]
\textbf{Probability for 1 Correct Answer: }\ \[ P(X = 1) = \binom{8}{1} (0.20)^1 (0.80)^7 = 0.3355 \]
\textbf{Probability for 2 Correct Answers: } \ \[ P(X = 2) = \binom{8}{2} (0.20)^2 (0.80)^6 = 0.2936 \]
Finally, we sum up these individual probabilities to find the total probability of getting fewer than 3 correct answers: \[ P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.1678 + 0.3355 + 0.2936 = 0.7969 \]
Therefore, the probability of getting fewer than 3 correct answers is 0.7969 or 79.69%.