To determine which plan has the higher payout in the long run, we need to calculate the expected value (EV) of the payout for each plan. The expected value is a weighted average of the possible payouts, with each payout being weighted by its probability.

The formula for expected value is:

$EV = \sum (Payout \times P(Payout))$

For Plan A:

$EV_A = (-10,000 \times 0.1) + (35,000 \times 0.36) + (95,000 \times 0.54)$

$EV_A = (-1,000) + (12,600) + (51,300)$

$EV_A = 62,900$

For Plan B:

$EV_B = (-40,000 \times 0.12) + (10,000 \times 0.69) + (100,000 \times 0.19)$

$EV_B = (-4,800) + (6,900) + (19,000)$

$EV_B = 21,100$

Conclusion:

• Plan A has an expected value of $62,900.
• Plan B has an expected value of $21,100.

Therefore, Plan A has the higher payout in the long run.

Would you like more details on how the expected value is calculated, or do you have any questions about the concept?

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Related Questions:

1. How do you calculate the expected value of a distribution?
2. What is the difference between expected value and variance?
3. How can the expected value be used in decision-making?
4. What does it mean if a payout has a negative expected value?
5. How does risk affect expected value calculations?

Tip: The expected value is a great way to compare options when dealing with uncertain outcomes. It gives you an "average" outcome based on the probabilities.