Sweet treats and probability

I love baking. More importantly, I love sweet things. However, I can’t and/or shouldn’t eat a full batch of cookies, bars, or brownies.  I bake for my students instead.

I use homemade sweet treats as prizes for going above and beyond the course work. This semester in my Stat 101 class, the first person in each section to explain the solution to the “Birthday Problem” in front of the class without notes got a Triple Chocolate Espresso Brownie. I’ll give you the premise of the question and and then the recipe for these great, easy brownies.

The “Birthday Problem” is a fascinating  probability problem that can be answered with a few basic probability rules that you learn in introductory statistics. The problem statement is how many people do you need to have in a room such that there is at least a 50% chance of at least two people in the room having the same birthday? To answer this, we need to simplify our world a bit. Let’s ignore Leap Day and assume there is an equal chance of being born on each of the 365 days of the year (which is probably not true these days with modern maternity medical practices).

First off, we notice that it is easier to calculate the chance that you don’t have at least two people with the same birthday in the room. The first person in the room could have any of the 365 days for their birthday (365/365) and then the second person in the room has to have a birthday on one of the 364 days left (364/365) and then the third person in the room has a birthday on one of the 363 days that aren’t called for yet (363/365). If there are n people in a room, the probability of getting at least two people with the same birthday is


This probability surpasses 0.5 when n = 23. One of my students eloquently explained this in class and got a homemade Triple Chocolate Espresso Brownie. The brownie recipe that I used, adapted from Cooks Illustrated, is listed below.

Triple Chocolate Espresso Brownies 
5 oz. semisweet chocolate, chopped
2 oz. unsweetened chocolate, chopped
8 T. unsalted butter, cut into quarters
3 T cocoa powder
1.5 T instant espresso powder
3 large eggs
1 1/4 C. sugar
2 t. vanilla extract
1/2 t. salt
1 C. all-purpose flour

1. Heat oven to 350 with oven rack in lower-middle position. Spray 8-inch square pan with cooking spray. Take one 12-inch piece of foil, fold it and lay it horizontally and press it into the corners. Take a second 12-inch piece of foil, fold it, and lay it vertically on the pan (making an cross with the other piece) and press it into the corners.  Spray foil with cooking spray.

2 . In medium saucepan, melt chocolates and butter on low, stirring frequently until mixture is smooth. Take off heat. Whisk in cocoa and espresso until smooth. Set aside to cool.

3. Whisk together eggs, sugar, vanilla, and salt in medium bowl until combined, about 15 seconds. Whisk warm chocolate mixture into egg mixture; then fold in flour until just combined. Pour mixture into prepared pan, spread into corners. Bake until slightly puffed and toothpick inserted in center comes out with a small amount of sticky crumbs clinging to it, 35-40 minutes.

4. Cool pan on wire rack to room temperature, about 2 hours, and then remove brownies from pan using foil handles. When ready to serve, cut into 1-inch squares. Enjoy!