Question

the mean is also called the expected value. if you rolled the lopsided die many times, youd expect the average of all the rolls to be about 2.1. as the table shows, the die is more likely to roll a 1 or 2. in 25 words or fewer, what would be the expected value for a fair 4 - sided die?

Explanation:

Step1: Calculate the sum of possible outcomes

For a fair 4 - sided die, the possible outcomes are \(1\), \(2\), \(3\), \(4\). The sum is \(1 + 2+3 + 4=\sum_{i = 1}^{4}i=\frac{4\times(4 + 1)}{2}=10\) (using the formula \(\sum_{i=1}^{n}i=\frac{n(n + 1)}{2}\) with \(n = 4\)).

Step2: Calculate the expected value

Since each outcome has a probability of \(\frac{1}{4}\), the expected value \(E(X)=\sum_{i = 1}^{4}x_ip_i\). Here \(x_i\) are the outcomes (\(1\), \(2\), \(3\), \(4\)) and \(p_i=\frac{1}{4}\) for \(i = 1,2,3,4\). So \(E(X)=\frac{1+2 + 3+4}{4}\) (because \(E(X)=\sum_{i=1}^{4}x_i\times\frac{1}{4}=\frac{\sum_{i = 1}^{4}x_i}{4}\)). Substituting \(\sum_{i=1}^{4}x_i = 10\), we get \(E(X)=\frac{10}{4}=2.5\).

Answer:

\(2.5\)