Please provide me handwritten solution

Problem $1$

You are serving as the only cashier at a supermarket. But there's no queue: instead, after you finish processing a customer, you decide which customer to process next.

Every minute, a customer arrives at your checkout: $10\%$ of the time, they have a big basket, which takes $5$ minutes to process. Otherwise, they have a small basket, which takes $30$ seconds to process.

Can you come up with a strategy that processes customers both efficiently and fairly? Feel free to create and solve your own extensions and variants!