MHB What is the expected outcome of a game of chance with varying probabilities?

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In a game of chance with a 30% chance of winning $5, a 50% chance of losing $4, and a 20% chance of breaking even, the expected profit can be calculated using the formula for expected value. The expected profit is determined by multiplying each outcome by its probability: 5 times 0.3 for winning, and -4 times 0.5 for losing, resulting in an expected profit of -$0.50. This indicates a projected loss of $0.50. The calculation can also be framed as expected loss, yielding the same result, demonstrating that the definitions of the random variable do not affect the outcome. Understanding these probabilities is crucial for assessing the expected financial results of such games.
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I have one more probability problem for you. I just don't know where to begin with this one.

"In a game of chance you have a 30% chance of winning $\$5$, a 50% chance of losing $\$4$ and a 20% chance of breaking even. What is your expected profit or loss?"
 
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So you need to take each outcome (positive = winning, negative = losing), multiply each one by its probability, and add it all up.
 
Let me get this straight:

I multiply 5 times 30 percent or 0.3
I also multiply 4 times 50% or 0.5
Then I add the sums of these two correct?

What do I do with the 20%?
 
Break even means no profit nor loss. Let $X$ be your profit, then $X$ is a random variable defined as (watch the signs!):$$ X = \left \{ \begin{array}{lll} 5, \quad & p = 0.3 \\ -4, \quad & p = 0.5 \\ 0, \quad & p = 0.2 \end{array} \right. $$Then the expected profit is given by $\mathbb{E}[X] = 5(0.3)+(-4)(0.5) = - 0.5$. This means that you expect a negative profit or hence a loss of $0.5 \$ $.

On the other hand, it's completely similar to say, let $Y$ be the loss, then $Y$ is a random variable defined as:$$ Y = \left \{ \begin{array}{lll} -5, \quad & p = 0.3 \\ 4, \quad & p = 0.5 \\ 0, \quad & p = 0.2 \end{array} \right. $$The expected loss is given by $\mathbb{E}[Y] = -5(0.3)+4(0.5) = 0.5$. Hence your expected loss is $0.5 \$$ which is exactly the same answer as above. It does not matter how you define the r.v as a loss of profit as long as you understand that a negative profit is a loss and vice versa.
 
Thanks Siron, this really helped break down everything for me! Thank you Ackbach for your assistance as well!
 
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