https://larspsyll.wordpress.com/2024/07/14/financial-economics-science-or-chance/

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To amend the Python program to reflect defined risk financial trades, hedging, and the existence of ETFs, you can simulate different trading strategies with varying probabilities of success. Here's an enhanced version of the script:

```python
import math

def binomial_coefficient(n, k):
    return math.factorial(n) // (math.factorial(k) * math.factorial(n - k))

def probability_single_person(n, k, p):
    probability = 0
    for i in range(k, n + 1):
        probability += binomial_coefficient(n, i) * (p ** i) * ((1 - p) ** (n - i))
    return probability

def probability_at_least_one_person(num_people, flips_per_person, min_heads, p):
    single_prob = probability_single_person(flips_per_person, min_heads, p)
    return 1 - (1 - single_prob) ** num_people

# Parameters for different strategies
num_people = 100 # number of people
flips_per_person = 10 # number of coin flips per person
min_heads = 9 # minimum number of heads for success

# Probability for a fair coin (pure chance)
p_fair = 0.5
probability_fair = probability_at_least_one_person(num_people, flips_per_person, min_heads, p_fair)
print(f"Probability with fair coin: {probability_fair:.4f} or {probability_fair * 100:.2f}%")

# Probability for a skilled trader with better odds (e.g., 60% success rate)
p_skilled = 0.6
probability_skilled = probability_at_least_one_person(num_people, flips_per_person, min_heads, p_skilled)
print(f"Probability with skilled trader: {probability_skilled:.4f} or {probability_skilled * 100:.2f}%")

# Probability for a hedged strategy (e.g., 70% success rate)
p_hedged = 0.7
probability_hedged = probability_at_least_one_person(num_people, flips_per_person, min_heads, p_hedged)
print(f"Probability with hedged strategy: {probability_hedged:.4f} or {probability_hedged * 100:.2f}%")

# Probability for an ETF with diversified risk (e.g., 55% success rate)
p_etf = 0.55
probability_etf = probability_at_least_one_person(num_people, flips_per_person, min_heads, p_etf)
print(f"Probability with ETF: {probability_etf:.4f} or {probability_etf * 100:.2f}%")
```

This enhanced script includes:

1. **Pure Chance (Fair Coin)**: Reflects random market behavior.
2. **Skilled Trader**: Simulates a higher probability of success due to skill.
3. **Hedged Strategy**: Reflects reduced risk and higher probability of success through hedging.
4. **ETFs**: Represents a diversified investment strategy with moderate success probability.

Adjust the probabilities (`p_skilled`, `p_hedged`, `p_etf`) based on the perceived effectiveness of each strategy.

--
bash-3.2$ python3 fin1.py
The probability that at least one person out of 100 gets heads at least 9 times is approximately 0.6604 or 66.04%
bash-3.2$ python3 fin2.py
Probability with fair coin: 0.6604 or 66.04%
Probability with skilled trader: 0.9913 or 99.13%
Probability with hedged strategy: 1.0000 or 100.00%
Probability with ETF: 0.9049 or 90.49%
bash-3.2$