Consider the following probability model where $ Y $ depicts the different prizes for another lottery in dollars and the corresponding probabilities:

\[
\begin{array}{c|cccc}
X = x & 0 & 10 & 50 & 100 \\
\hline
P(X = x) & 0.4 & 0.3 & 0.2 & 0.1 \\
\end{array}
\]

What is the expected (mean) value of $ Y $?

The expected value (mean) of the given lottery game, represented by Y, is $20. This is calculated by multiplying each prize amount by its corresponding probability and summing the results. This indicates the average prize a player could expect to win.

Explanation:

The subject of this question is mathematics, specifically probability and statistics. The question is asking for the expected (mean) value of a probability model for a lottery game, represented by Y. The expected value can be calculated using the formula E(X) = µ = Σ xP(x), where each value of the random variable (x) is multiplied by its corresponding probability (P[x]) and the products are summed.

So, applying the information given to the formula, we have the expected value E(X) = (0*0.4) + (10*0.3) + (50*0.2) + (100*0.1) = 20. This means, on average, a player would expect to win $20 in this lottery game.

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