Question

a person randomly selects one of six envelopes. each envelope contains a check that the person gets to keep. determine the persons expectation if the six envelopes contain checks for $126, $235, $327, $446, $496, and $1093. the expected value is $ (round to the nearest cent as needed.)

Explanation:

Step1: Recall the formula for expected value

The formula for the expected value \(E(X)\) of a discrete random variable is \(E(X)=\sum_{i = 1}^{n}x_ip_i\). In the case of equally - likely outcomes (here, each envelope has an equal probability of being selected, \(p_i=\frac{1}{6}\) for \(i = 1,2,\cdots,6\) and \(x_i\) are the values of the checks).

Step2: Calculate the sum of the values of the checks

Let \(x_1 = 126\), \(x_2=235\), \(x_3 = 327\), \(x_4=446\), \(x_5 = 496\), \(x_6=1093\). Then \(\sum_{i=1}^{6}x_i=126 + 235+327+446+496+1093\).

$$
LATEXBLOCK0
$$

Step3: Calculate the expected value

Since \(p_i=\frac{1}{6}\) for \(i = 1,\cdots,6\), \(E(X)=\sum_{i = 1}^{6}x_ip_i=\frac{1}{6}\sum_{i = 1}^{6}x_i\). Substitute \(\sum_{i = 1}^{6}x_i = 2723\) into the formula: \(E(X)=\frac{2723}{6}\approx453.83\)

Answer:

\(453.83\)