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. (12) = 2 how many of the outcomes do you expect to result in a number less than 6? 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: Calculate probability of rolling a 1

The probability of rolling a 1 on a fair - numbered cube is $p=\frac{1}{6}$ since there is 1 favorable outcome (rolling a 1) out of 6 possible outcomes (numbers 1 - 6).

Step2: Use expected - value formula

The expected - value formula for a binomial situation is $E(X)=np$, where $n$ is the number of trials and $p$ is the probability of success on a single trial. Here, $n = 12$ (number of rolls) and $p=\frac{1}{6}$. So, $E(X)=12\times\frac{1}{6}$.

Step3: Compute the result

$12\times\frac{1}{6}=2$.

for second part:

Step1: Calculate probability of rolling a number less than 7

The numbers on the cube are 1 - 6, so all 6 outcomes are less than 7. The probability of rolling a number less than 7 on a single roll is $p = 1$ since all 6 possible outcomes satisfy the condition.

Step2: Use expected - value formula

Using $E(X)=np$ with $n = 12$ (number of rolls) and $p = 1$, we get $E(X)=12\times1$.

Step3: Compute the result

$12\times1=12$.

Answer:

2