**Python Puzzles**

This problem is called the coupon collector’s problem.

A cereal is running a promotion where if you collect all 10 coupons, you win a prize. A coupon is only available if you buy a box of their cereal. What is the average number of box purchases needed in order to collect all 10 coupons?

When I run the following simulation, I get the following:

```
Minimum number of box purchases needed: 10
Maximum number of box purchases needed: 134
Average number of box purchases needed: 29.21431
```

The Python code below also includes a histogram to show the distribution of box purchases needed to collect all 10 coupons.

import random import matplotlib.pyplot as plt def collect_coupons(): coupons = set() num_purchases = 0 while len(coupons) < 10: coupon = random.randint(1, 10) coupons.add(coupon) num_purchases += 1 return num_purchases num_simulations = 100000 total_purchases = 0 purchases_list = [] for i in range(num_simulations): purchases = collect_coupons() total_purchases += purchases purchases_list.append(purchases) avg_purchases = total_purchases / num_simulations print("Average number of box purchases needed:", avg_purchases) plt.hist(purchases_list, bins=range(1, max(purchases_list)+1), align='left') plt.xticks(range(1, max(purchases_list)+1, 20)) # show every 20th number plt.xlabel('Number of box purchases') plt.ylabel('Frequency') plt.title('Distribution of box purchases needed to collect all 10 coupons') plt.show() min_purchases = min(purchases_list) max_purchases = max(purchases_list) avg_purchases = total_purchases / num_simulations print("Minimum number of box purchases needed:", min_purchases) print("Maximum number of box purchases needed:", max_purchases) print("Average number of box purchases needed:", avg_purchases)