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dmitriy555 [2]
3 years ago
14

Suppose someone gives you 8 to 2 odds that you cannot roll two even numbers with the roll of two fair dice. This means you win ​

$ if you succeed and you lose ​$ if you fail. What is the expected value of this game to​ you? Should you expect to win or lose the expected value in the first​ game? What can you expect if you play ​times? Explain.
Mathematics
1 answer:
krek1111 [17]3 years ago
6 0

Complete question :

Suppose someone gives you 8 to 2 odds that you cannot roll two even numbers with the roll of two fair dice. This means you win $8 if you succeed and you lose $2 if you fail. What is the expected value of this game to​ you? What can you expect if you play 100 times.

Answer:

$0.5 ; win $50 with 100 rolls

Step-by-step explanation:

From a roll of two fair dice; probability of obtaining an even number :

Even numbers = (2, 4, 6) = 3

P = 3 /6 = 1 /2

For 2 fair dice ; probability of rolling two even numbers : independent event.

1/2 * 1/2 = 1/4

Hence, p(success) = 1/4 ; P(failure) = 1 - 1/4 = 3/4

Probability table

Winning = $8 or loss = - $2

X : ____ 8 ______ - 2

P(x) __ 1/4 ______ 3/4

Expected value : E(x) = ΣX*P(x)

E(x) = (8 * 1/4) + (-2 * 3/4)

E(x) = 2 - 1.5

E(x) = $0.5

Since expected value is positive, the expect to win

If played 100 times;

Expected value = 100 * $0.5 = $50

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The right answer is c. –2.24, 2, 2.24


This question needs to be solved in two ways. First, using a graphing calculator. Next, using a system of equations.


1. Using a graphing calculator.


We have the following polynomial equation:

x^3-5x+5=2x^2-5


By ordering this equation we have:

x^3-2x^2-5x+10=0


So, we can say that this equation comes from a function given by:

f(x)=x^3-2x^2-5x+10


Thus, by plotting this function, we have that the graph of this function is indicated in Figure 1. By zooming, we can see, in Figure 2, that the roots of the polynomial equation are the x-intercepts of f(x) which are:


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Finally, rounding noninteger roots to the nearest hundredth we have:


\boxed{Root_{1}=-2.24} \\ \\ \boxed{Root_{2}=2} \\ \\ \boxed{Root_{3}=2.24}


2. Using a system of equations.


The ordered equation is:

x^3-2x^2-5x+10=0


By arranging to factor out we have:

x^3-5x-2x^2+10=0


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