91Ó°ÊÓ

A probability distribution describes how the values of a random variable are distributed. In a more simplistic term, it tells us how likely each outcome is to occur. In the exercise provided, we deal with a **Binomial Distribution**. This is a specific type of probability distribution often used when there are two possible outcomes. For instance, every question on the exam has five choices, but only one is correct; hence it is treated as a success (correct) or failure (incorrect). When you guess on each question, there are 100 independent trials and therefore, this can be defined as a Binomial Distribution with 100 trials, where each trial has a success probability of 1/5. The probability distribution allows us to compute important metrics like the expected value and standard deviation, helping us understand more about what to expect from this random guessing scenario.

The **Expected Value** is a concept representing the average or mean of all possible outcomes. In probability, it gives us an idea of what the average result would be if we were to repeat the random process infinite times. In a binomial distribution, the expected value is calculated with the formula:

• \( \mu = n \times p \)

where \( n \) is the number of trials (here, 100 questions) and \( p \) is the probability of success in each trial (guessing a question right, which is 1/5). Simplifying it gives:

• \( \mu = 100 \times (1/5) = 20 \)

This means, if you guess randomly throughout the exam, on average, you should expect to get 20 questions correct. It's a way to prepare what you might reasonably achieve if you were to repeat the process many times.

**Variance** and **Standard Deviation** measure the spread of the probability distribution. Variance tells us how much the possible outcomes deviate from the expected value, and standard deviation is the square root of the variance. For a binomial distribution, the variance (\( \sigma^2 \)) is calculated using:

• \( \sigma^2 = n \times p \times (1 - p) \)

Given the problem illustrates a guess for each of the 100 questions with a probability of success of 1/5, we have:

• \( \sigma^2 = 100 \times (1/5) \times (4/5) = 16 \)

The standard deviation is simply:

• \( \sigma = \sqrt{16} = 4 \)

This suggests that most of the time, your score will deviate by about 4 questions from the expected score of 20. Therefore, it's quite unlikely that you'd score much higher, for example, over 50, as it's significantly beyond just one standard deviation from 20.