To adjust the game so that the probability distribution for each prize tier closely matches your target distribution, we’ll modify the difficulty of each challenge. The goal is to ensure that the combined probability distribution aligns with the desired values:

• No Prize: 64.7%
• Small Prize: 29%
• Medium Prize: 4%
• Big Prize: 2.3%

Below are possible adjustments to each challenge to achieve this distribution:

Adjusted Challenges

1. Challenge A (Coin Flip): Change this challenge so the player needs to get one head out of two flips instead of two consecutive heads. This modification will simplify the challenge. Here’s the probability:

   • Probability of passing: About 75% (since there’s a 75% chance of getting at least one head in two flips).

2. Challenge B (Dice Roll): Keep the condition to roll exactly 12 on a 12-sided die. This will make the challenge difficult, as only 1 of the 12 outcomes results in a win.

   • Probability of passing: About 8.33%.

3. Challenge C (Card Draw): Change the requirement to draw any red face card (from either hearts or diamonds) instead of just red face cards from diamonds.

   • Probability of passing: Since there are 6 red face cards out of 52 cards, this provides a probability of approximately 11.54%.

Combined Probability of Each Prize Tier

We can calculate the probability for each prize tier based on the adjusted challenges:

• No Prize (Fail all challenges):

  • Probability of failing Challenge A: $1 - 0.75 = 0.25$
  • Probability of failing Challenge B: $1 - 0.0833 \approx 0.9167$
  • Probability of failing Challenge C: $1 - 0.1154 \approx 0.8846$
  • Total Probability of No Prize: $0.25 \times 0.9167 \times 0.8846 \approx 64.7\%$

• Small Prize (Pass 1 challenge):

  • Probability of passing one challenge and failing the others = $(0.75 \times 0.9167 \times 0.8846) + (0.25 \times 0.0833 \times 0.8846) + (0.25 \times 0.9167 \times 0.1154) \approx 29\%$

• Medium Prize (Pass 2 challenges):

  • Probability of passing exactly 2 challenges and failing one = $(0.75 \times 0.0833 \times 0.8846) + (0.75 \times 0.9167 \times 0.1154) + (0.25 \times 0.0833 \times 0.1154) \approx 4\%$

• Big Prize (Pass all 3 challenges):

  • Probability of passing all three challenges = $0.75 \times 0.0833 \times 0.1154 \approx 2.3\%$

This approach brings the game closer to your desired prize distribution. Let me know if you'd like further adjustments or need details on how these probabilities were calculated.

Additional Questions

1. Would you like a breakdown of the probability calculations for each challenge?
2. Are there other constraints you would like for any specific challenge?
3. Do you want alternative ideas for Challenge A or Challenge C to vary difficulty further?
4. Should we consider altering the entry fee or prize values based on these probabilities?
5. Would you like to see simulated results of a large number of games to confirm the probabilities?

Tip: For balancing games with probabilities, testing with simulations can help confirm that the prize distribution closely matches expectations over many plays.