# How to express the relationship between two variables based of their expected value

A biased coin has probability 0.6 of turning up heads. You win $$x$$ dollars if a head comes up and you lose $$y$$ dollars if a tail comes up. If your expected winnings is $$0$$ dollars, what is the relationship between x and y?

So far I have:

$$P(x) = 0.6$$

$$P(y) = 0.4$$

$$E[x] =$$expected winnings$$= 0$$

$$E[x] = x(0.6) + y(0.4) = 0$$

$$x = -y(0.4)/(0.6)$$

I'm kind of stumped on how to continue from here, how can I use the information given to express the relationship between x and y?

• You correctly got $0.6x + 0.4y = 0$ which leads to $x=-\frac{2}{3} y$ as you got. That is the final answer. There is nothing more to do, this equation describes the relationship between $x$ and $y$ in full detail. Any alternate ways of expressing this information is just that, an alternate way of expressing the same information. – JMoravitz Feb 25 at 17:31
• @JMoravitz I think it is not correct. – callculus Feb 27 at 16:55
• @callculus how so? Just the difference in the negative sign? That merely varies based on whether you consider a positive value of y to refer to a loss or whether a positive value of y to refer to a gain. Both are valid interpretations and ways of setting up the problem that have little to no impact on the final results beyond a sign change. – JMoravitz Feb 27 at 20:03

You´re on the right track, but you haven´t taken into account that you $$\texttt{lose}$$ $$y$$ dollars if a tail comes up. Therefore the equation is $$0.6\cdot x-0.4\cdot y=0$$
Now solve the equation for $$\frac{x}y$$.
You want to look at the ratio $$\frac {x}{y}$$.
If $$x = -y (0.4) / (0.6)$$, then $$x / y = - (0.4) / (0.6)$$
$$\frac {x}{y} = - \frac {2}{3}$$