Ensuring Diversity: Minimizing Jelly Babies to Get Each Flavor

Tom found himself with a diverse selection of jelly babies in his pocket—8 watermelon, 4 vanilla, and 4 butter popcorn varieties. The question arises: what is the smallest number of jelly babies Tom must take out to be certain that he has at least one from each flavor? This puzzle might seem simple at first glance, but by diving into the concept of the worst-case scenario, we can unravel the solution.

Understanding the Worst-Case Scenario

The key to solving this problem is to consider the worst possible sequence of draws. In the worst-case scenario, Tom could draw all of the jelly babies from two of the flavors first. To guarantee that Tom gets at least one jelly baby of each flavor, we need to account for this worst-case situation.

Worst-Case Analysis

Let's break down the worst-case scenario step-by-step:

Watermelon Jelly Babies: Tom could draw all 8 watermelon jelly babies. Vanilla Jelly Babies: After the watermelon jelly babies, Tom could draw all 4 vanilla jelly babies.

At this point, Tom has drawn 12 jelly babies in total—8 watermelon and 4 vanilla. However, he still does not have any butter popcorn jelly babies in his hand. To ensure he has at least one of each flavor, Tom must draw one more jelly baby.

The total number of jelly babies Tom needs to take out to guarantee that he has at least one from each flavor is:

8 watermelon 4 vanilla 1 butter popcorn

Therefore, the sum is:

8 4 1 13

To be certain, Tom must take out 13 jelly babies.

Minimum Number of Jelly Babies to Ensure Diversity

The analysis above confirms that the smallest number of jelly babies Tom must take out to guarantee he has at least one of each flavor is 13. However, it is important to note that the minimum number could potentially be lower depending on the initial draw.

For instance, if Tom were to draw a butter popcorn jelly baby first, he would only need to take out an additional 12 jelly babies (8 watermelon 4 vanilla 1 butter popcorn) to ensure he has at least one of each flavor. This scenario indicates that the actual minimum number could be as few as 3, but to be absolutely certain, the worst-case scenario dictates 13.

By understanding the worst-case scenario, we can ensure that we account for the most unfavorable outcome, guaranteeing that Tom always has at least one of each flavor of jelly babies.