Counting Tigers in the World’s Richest Math Competition

A probability question from the 2022 Alibaba Global Mathematics Competition

The Competition

Want to win USD $40,000? That was the prize money awarded to each of the four gold medallists in the 2022 Alibaba Global Math Competition. The qualifying round of the competition is open to anyone. I should warn you though, that medallists tend to be professional mathematicians and/or former IMO medallists!

The question presented in this article was from the 2022 qualifying round. Programming is allowed, which makes it easier for amatuer enthusiasts like myself to get a foot in the door. I hope you’ll agree it’s a lovely problem…

The Problem

A milk company runs a Chinese New Year promotion, giving away a red packet with each milk carton. Each packet has one of the characters

• 虎 (hǔ — tiger)
• 生 (shēng — gain)
• 威 (wēi — strength)

, with equal probability. People try to collect two 虎 (hǔ), one 生 (shēng) and one 威 (wēi) so they can form the lucky phrase “虎虎生威”, which is a good blessing for 2022, the Year of the Tiger.

What is the expected number of milk cartons required to obtain “虎虎生威”, i.e. two 虎 (hǔ), one 生 (shēng) and one 威 (wēi) in any order?