Question

Nicole is playing a game of chance in which she rolls a number cube with sides numbered from 1 to 6

Answer 1

GIVEN;

We are told that Nicole plays a game in which a fair die with faces numbered 1 to 6 appears for each role. Each number appears randomly.

Required;

To determine the expected value of playing this game.

Step-by-step solution;

We can calculate the expected value of playing the game by multiplying the value of each outcome by its probability of occurrence and then and then we add up all the results.

Note that for a fair die, the probability of each number on a roll is 1/6. That is, each number has the same chance (1 out of 6) of occurrence.

The probability distribution is;

`P[1]=(1)/(6),P[2]=(1)/(6),P[3]=(1)/(6),P[4]=(1)/(6),P[5]=(1)/(6),P[6]=(1)/(6)`

The probability of rolling a 4, 5 or 6 is the addition of probabilities and that is;

`\begin{gathered} P[4]\text{ }or\text{ }P[5]\text{ }or\text{ }P[6]=(1)/(6)+(1)/(6)+(1)/(6)=(3)/(6) \\ \\ =(1)/(2) \end{gathered}`

We can now calculate the expected value as follows;

`\begin{gathered} EV=[1*(1)/(6)]+[2*(1)/(6)]+[3*(1)/(6)]-[2*(1)/(2)] \\ \\ EV=(1)/(6)+(1)/(3)+(1)/(2)-1 \\ \\ EV=1-1 \\ \\ EV=0 \end{gathered}`

The expected value of playing the game is $0

After playing the game many times, Nicole can expect to break even (neither gain nor lose money)

ANSWER:

(A) The expected value of playing the game is $0

(B) After playing the game many times, Nicole can expect to break even (neither gain nor lose money)