GIVEN:
You pay $5 to play a game. To play the game you spin a spinner with 3 colors.
If the spinner lands on blue you earn $20. If the spinner lands on green, you get your $5 back. If the spinner lands on red, you loose your money.
The probabilities of the spinner landing on each color is given in the chart below.
[tex]\begin{gathered} Color----------Probabilities \\ \\ Blue-----------0.19 \\ \\ Green-----------0.14 \\ \\ Red------------0.67 \end{gathered}[/tex]Required;
What is the expected value of this game, given to the nearest penny?
Step-by-step solution;
To solve the question, note that the probability distribution has a blank space. We have the probabilities of landing on a blue color and on a green color. The probability of an event is usually between 0 and 1.00. Therefore, for an experiment with 3 outcomes, the probabilities would all be equal to 1 (regardless of the value given to each outcome). Hence, we are able to calculate the probability of landing on a red color as;
[tex]\begin{gathered} Red=1-(0.19+0.14) \\ \\ Red=0.67 \end{gathered}[/tex]To solve for the expected value of this event, we now have to multiply each probability distribution by the reward attached to each outcome/probability.
For landing on a blue;
[tex]\begin{gathered} Expected\text{ }value=P(x)\times x \\ \\ Expected\text{ }value=0.19\times(\text{\$}20-\text{\$}5) \\ \\ Expected\text{ }value=0.19\times15 \\ \\ EV=2.85 \end{gathered}[/tex]For landing on a green;
[tex]\begin{gathered} EV=0.14\times(\text{\$5}-\text{\$5\rparen} \\ \\ EV=0.14\times0 \\ \\ EV=0 \end{gathered}[/tex]For landing red;
[tex]\begin{gathered} EV=0.67\times(\text{\$0}-\text{\$}5) \\ \\ EV=0.67\times(-5) \\ \\ EV=−3.35 \end{gathered}[/tex]Now we can calculate the expected earnings from playing this game.
We sum up the individual expected earnings as follows;
[tex]\begin{gathered} Expected\text{ }earnings=\Sigma[xP(x)] \\ \\ Expected\text{ }earnings=2.85+0+(-3.35) \\ \\ Expected\text{ }earnings=−0.5 \end{gathered}[/tex]We now have a negative value which means based on the conditions given, the expected earnings from playing this game is a loss of $0.50
ANSWER:
Expected value
[tex]Expecetd\text{ }value=\text{\$}-0.50[/tex]