1. Probability ($P$): Always a number between 0 and 1.
2. Expected Value ($E$): Outcome $ imes$ Probability.
3. Independence: Does the first event "reset the coin"?
4. Sum of One: All probabilities in a set must add up to 1.0 (100%).
Using a standard 6-sided die.
The Single Roll: What is the probability of rolling an even number (2, 4, 6)?
The Double Roll: What is the probability of rolling a "6" and then another "6"? (Independent events).
The Sum: If you roll two dice, how many ways can you get a sum of 7? What is the probability?
If you roll a "1" three times in a row... are you "less likely" to roll a 1 on the fourth try? Explain why.
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Calculate the $E[X]$ for these scenarios.
The Small Seed: A crop has a 70% chance of yielding 100 units and a 30% chance of yielding only 10 units. What is the expected yield?
The Talent Trade: You have a choice between:
A) A guaranteed gift of $\$10$.
B) A 10% chance of winning $\$200$ and a 90% chance of getting $\$0$.
Which choice has the higher Expected Value?
The Bag of Stones: A bag has 5 White stones and 5 Black stones.
1. You pull out a White stone and keep it.
2. What is the probability that the second stone you pull is also White?
The village must choose a project:
1. **Repair the Bridge:** Cost $\$500$. 100% chance of saving $\$800$ in trade fees.
2. **Dig a Mine:** Cost $\$1,000$. 20% chance of finding gold worth $\$10,000$. 80% chance of finding nothing.
Task: Calculate the "Net Expected Value" (Expected Gain - Cost) for both. Which one is the better "Prophetic Stewardship"?
Objective: Explain Expected Value using candies to a younger student.
The Activity:
1. Offer them 1 candy for sure.
2. Offer them a coin flip: Heads = 3 candies, Tails = 0 candies.
The Question: "If we play this game 10 times, which way will give you the most candy?"
The Lesson: "Sometimes taking a risk is smarter than playing it safe, if the reward is big enough. God wants us to use our brains to find the best way to grow."
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