Intuitive explanation of the expected value In regards to the expected value topic an example I have read is:

If the probability of an event happening is $\frac{1}{5000}$ and the
payoff if it occurs is \$$200,000$ then by multiplying
the probability with the amount that is expected to be payed out for each such occurrence happening (think of insurance policies)

I am not sure how to intuitively interpret this multiplication. I guess if the probability is $100\%$ then the payout is \$$200,000$ and if it is $50\%$ it is \$$100,000$ but seems to me kind of arbitrary/crude. Any help here?
 A: Since this is an interpretation question, I feel free to answer,
without work being shown.
Go back to the original question.
Suppose that $5000$ people form a pool and agree to evenly share the proceeds.
On average, exactly one of the $5000$ people will win $\$200,000,$ while
everyone else wins nothing.
This money will then be evenly divided by everyone in the pool.
Each person will therefore get $\frac{\$200,000}{5000}.$
This represents each person's expected value.

Intuitively, the same result can be computed by assuming that 1 person, operating alone, enters a contest $5000$ times.  Since he expects to win, on average, once out of every $5000$ times, he expects to win once.
This means that as a result of his entering the contest $5000$ times, on average, he (somewhere along the way) wins the contest exactly once, receiving $\$200,000.$
This means that his expectation is that it takes $5000$ occurrences to win $\$200,000.$  Therefore, his expectation is that his average win, for each occurrence is
$\frac{\$200,000}{5000}.$

Addendum
Responding to the two comment/questions of Jim:

In your first example I think that is equivalent to stating that the company pays out $200,000 for every 5,000 customers...

Yes, I agree.

from whom it received $5,000 \times \$50 = \$250,000.$ Hence the profit is $\$50,000$ for the company which means $\$10$ per customer.

First of all, nowhere in your original question is there any mention of customers paying $\$50$ for a policy.  However, taking this added constraint at face value, I completely agree with your math.  That is, the company's expected profit per customer is
$$\frac{(\$50 \times 5000) - (\$200,000)}{5,000 ~\text{customers}} = 
\frac{\$50,000}{5,000 ~\text{customers}}
= \frac{\$10}{1 ~\text{customer}}.
$$

Somehow the way you describe it I can not directly understand it except how I interpreted it, because each person in your example literary will not get $\$40$ so it can't be the expected value. The company can expect to get $\$10$ per person and that would be the actual expected value. Am I confused here?

My original answer reflected in effect that there was no stated cost for the prize, which in the insurance company analogy, translates into free insurance policies.  Once you add the constraint that each policy costs $\$50$, then yes, each customer's expectation is reduced by $\$50$.
This means that each customer's expectation changes from $\$(+40)$ to $\$(-10)$, which exactly corresponds to the insurance company's profit of $\dfrac{\$10}{1 ~\text{customer}}$.

In your second analogy, I think you mean that either we state the person wins $\$200,000$ once or $\$40$ per participation on average are equivalent definitions. Isn't the cost of $\$50$ per participation required to be included in the second analogy?

Yes, the cost of $\$50$ per participation is required to be included.  However, in my original answer, I could not include it, because in the original problem there was no specified cost per participation.
