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Find the expected value of the winnings from a game that has the following payout probability distribution:

\begin{tabular}{c|ccccc}
Payout [tex]$(\$[/tex])[tex]$ & 0 & 2 & 5 & 6 & 10 \\
\hline
Probability & 0.57 & 0.21 & 0.04 & 0.04 & 0.14
\end{tabular}

Expected Value $[/tex]=[tex]$ $[/tex]\square$

Round to the nearest hundredth.

Sagot :

To find the expected value of the winnings from the given game, we need to use the concept of expected value in probability theory. The expected value (EV) of a random variable is a measure of the center of its probability distribution and is calculated by summing the products of each outcome and its corresponding probability.

Here is the step-by-step solution:

1. Identify the payouts and probabilities:
[tex]\[ \begin{aligned} &\text{Payouts}:\quad &&0\, \$,\, 2\, \$,\, 5\, \$,\, 6\, \$,\, 10\, \$ \\ &\text{Probabilities}:\quad &&0.57,\, 0.21,\, 0.04,\, 0.04,\, 0.14 \end{aligned} \][/tex]

2. Multiply each payout by its probability to find the contribution of each outcome to the expected value:

[tex]\[ \begin{aligned} &0 \times 0.57 = 0 \\ &2 \times 0.21 = 0.42 \\ &5 \times 0.04 = 0.20 \\ &6 \times 0.04 = 0.24 \\ &10 \times 0.14 = 1.40 \\ \end{aligned} \][/tex]

3. Sum these contributions to get the total expected value:

[tex]\[ \begin{aligned} EV &= 0 + 0.42 + 0.20 + 0.24 + 1.40 \\ &= 2.26 \end{aligned} \][/tex]

So, the expected value of the winnings from the game is [tex]\( 2.26 \)[/tex] dollars.

4. Round to the nearest hundredth where necessary:
In this case, [tex]\( 2.26 \)[/tex] is already rounded to the nearest hundredth.

Therefore, the expected value of the winnings is:
[tex]\[ \boxed{2.26} \][/tex]