Determine the likelihood of achieving the desired sum through various lottery drawings

If I were to participate in multiple lotteries with varying prize amounts and different odds of winning, how can I determine the likelihood of winning at least a specified value if I play each lottery simultaneously and only once?

  • Lottery A: Has a 50% chance of winning 5 gold.
  • Lottery B: Offers a 40% chance of winning 6 gold.
  • Lottery C: Provides a 30% chance of winning 7 gold.

I am interested in finding a method to calculate the probability of winning an amount equal to or greater than a set target value (such as 10 gold) when participating in all these lotteries simultaneously. This solution should be scalable to help me analyze around 40 different lotteries.

The input would consist of a list containing tuples representing the probability and prize size of each lottery, for example:

lottery_list = [(0.5, 5), (0.4, 6), (0.3, 7)]

Following this, a function could be utilized to compute the probability of winning at least the specified target value, such as 10 gold in this case:

prob = win_at_least(lottery_list, target_val=10)
python

Answer №1

After implementing a solution on my own, I was able to optimize the code by excluding irrelevant combinations of 'downstream' lotteries if the set of lotteries has already produced the target value. This enhancement significantly improved performance, reducing the processing time to about 10 ms for 40 lotteries while maintaining scalability.

import numpy as np
import pandas as pd


def calculate_lottery_odds(lottery_list: list, target: float) -> float:
    """
    Determines the probability of winning at least the target
    value from a list of lotteries with varying win probabilities and values.

    Args:
        lottery_list (list): List of tuples containing lottery
            probabilities and values.
        target (float): Target value to be achieved.

    Returns:
        float: Probability of winning at least the target value.
    """
    # Create a dataframe using the lottery list.
    df = pd.DataFrame(lottery_list, columns=["probability", "value"])

    # Sort the data frame in descending order of value.
    df_sorted = df.sort_values("value", ascending=False, ignore_index=True)

    probs = df_sorted["probability"].values
    values = df_sorted["value"].values

    length = len(df_sorted)
    mask = np.ones(length, dtype=int)

    total_odds = 0

    while True:
        odds = 1
        value = 0

        for idx, prob in enumerate(probs):
            if mask[idx] == 1:
                odds *= prob
                value += values[idx]
            else:
                odds *= 1 - prob

            if value >= target:
                total_odds += odds
                mask[idx] = 0
                break

            elif idx == length - 1:
                largest_active_idx = 0

                for i, v in enumerate(mask):
                    if v == 1:
                        largest_active_idx = i

                mask[largest_active_idx] = 0
                mask[largest_active_idx + 1 :] = 1

        if np.sum(mask) == 0:
            break

    return total_odds

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