Day 4 — Probability
Time: ~30 min · Topic: probability
The problem
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. You draw 2 marbles without replacement. What is the probability that both marbles are the same color?
Hint: Think about each color separately — what's the chance of drawing two reds, two blues, OR two greens?
Try it yourself
Take a few minutes. Don't peek at the steps. If you get stuck, the hint above is enough to nudge you forward.
Step-by-step solution
- Find P(both red) = (5/10) x (4/9) = 20/90
- Find P(both blue) = (3/10) x (2/9) = 6/90, and P(both green) = (2/10) x (1/9) = 2/90
- These outcomes are mutually exclusive, so add them: 20/90 + 6/90 + 2/90 = 28/90 = 14/45
Answer: 14/45 (approximately 31.1%)
Why this matters
DNA paternity tests use the exact same probability logic — calculating the chance that genetic markers match by coincidence rather than inheritance!
This day in math: The Man Who Bottled a New Geometry
April 24, 1849 · Felix Klein
On April 25, 1849, Felix Klein was born in Düsseldorf, Prussia. He became a full professor at the astonishing age of 23 and went on to revolutionize mathematics with his 1872 Erlangen Program, which unified all of geometry under the language of symmetry and group theory. He also introduced the Klein bottle — a mind-bending surface with no inside or outside that can only exist without self-intersection in four dimensions.
Why it matters: Klein's Erlangen Program fundamentally changed how mathematicians think about geometry, showing that every type of geometry is really a study of what stays the same under certain transformations. His ideas laid the groundwork for modern topology, theoretical physics, and the role of symmetry throughout all of mathematics.
Did you know?
Einstein never actually wrote E = mc^2 in his groundbreaking 1905 paper. He originally expressed the idea as m = L/c^2, using the letter L for energy, and framed it as mass decreasing when a body releases energy — the reverse of how we usually think about the equation today.
In his 1905 paper, Einstein used "L" for energy (a convention common in German physics at the time) and concluded that a body's mass diminishes by L/c^2 when it radiates energy. The now-iconic form E = mc^2 only appeared in Einstein's later work and was popularized over the following decades. The original framing — that energy loss reduces mass — was actually a more precise statement of his discovery than the version the world came to memorize.
Done?
- Solved the problem (or read through the steps)
- Read the history note
- Read the "did you know" fact