Question

benjamin rolls a number cube labeled with the numbers 1 - 6 a total of 12 times. determine the expected value and explain your solution method. sample problem how many of the outcomes do you expect to result in a 1? i would expect 2 out of the 12 outcomes to result in a 1. e(12)=2 how many of the outcomes do you expect to result in an even number? enter the answer in the space provided. use numbers instead of words. explain your solution method. enter the answer in the space provided.

Explanation:

Step1: Determine probability of rolling a 1

A fair - numbered cube has 6 faces. The probability \(p\) of rolling a 1 on a single roll is \(p=\frac{1}{6}\).

Step2: Use expected - value formula

The expected - value formula for the number of successes \(E(X)\) in \(n\) trials is \(E(X)=n\times p\), where \(n = 12\) (number of rolls) and \(p=\frac{1}{6}\). So \(E(X)=12\times\frac{1}{6}\).

Step3: Calculate the result

\(12\times\frac{1}{6}=2\).

For the second part:

Step1: Determine probability of rolling an even number

The even numbers on a cube are 2, 4, 6. So there are 3 even numbers out of 6. The probability \(q\) of rolling an even number on a single roll is \(q = \frac{3}{6}=\frac{1}{2}\).

Step2: Use expected - value formula

Using \(E(Y)=n\times q\) with \(n = 12\) and \(q=\frac{1}{2}\), we get \(E(Y)=12\times\frac{1}{2}\).

Step3: Calculate the result

\(12\times\frac{1}{2}=6\).

Answer:

The expected number of 1s in 12 rolls is 2. The expected number of even - numbered outcomes in 12 rolls is 6.