To give the probability distribution, we need to calculate the probability of each possible outcome and the value of this outcome.
We have 5000 raffle, 1 will win the first prize, 2 will win the second prize, 3 will win the third prize and the rest 4994 will win no prize.
The first prize is $500, but the raffle cost $1, so the outcome is actually $499.
The second prizes are $200 each, minus the cost we have an outcome of $199.
The third prizes are $100 each, minus the cost we have an outcome of $99.
The others will not receive prizes, but they will still have the cost of $1, so the outcome is -$1.
The first prize is 1 in 5000, so the probability is 1/5000
The second prizes are 2 in 500, so the probability is 2/5000
The third prizes are 3 in 5000, so the probability is 3/5000
The lost is the rest of the 4994 in 500, so the probability is 4994/5000
So, the table for the probability distributions is:
Value gained | P(x)
$499 | 1/5000
$199 | 2/5000
$99 | 3/5000
-$1 | 4994/5000
To calculate the expected value, we multiply the value by its probability and add them:
[tex]\begin{gathered} E(x)=499\cdot\frac{1}{5000}+199\cdot\frac{2}{5000}+99\cdot\frac{3}{5000}-1\cdot\frac{4994}{5000}_{} \\ E(x)=\frac{499}{5000}+\frac{398}{5000}+\frac{297}{5000}-\frac{4994}{5000} \\ E(x)=\frac{499+398+297-4994}{5000} \\ E(x)=-\frac{3800}{5000} \\ E(x)=-0.76 \end{gathered}[/tex]So, the expected value if -$0.76.
