Word Problem Wednesday: Keeping Your Marbles!

Here’s a classic use for math that’s useful at the races. Ready? 1, 2, 3 go!

There are 100 marbles in a bag. They’re numbered 0, 1, 2, all the way through 99. What is the probability that if you draw three marbles out of the bag at random without replacing them, the number on the first marble will be 0, the number on the second will be 1, and the number on the third will be 2?

Answer waaaay down below!

We've got to admit that marbles are pretty wonderful... wonderful, pretty, and oh so versatile! Affordable and plentiful, marbles are used as collectibles, decorations, games, toys, and for any purpose we can dream up... like this word problem :-) . Here's how they are made.

Did you get 1 out of 970,200?

Here is how we do it. The probability of drawing the 0 first is ¹/₁₀₀. If the marble isn’t replaced, then the probability of drawing the 1 next is ¹/₉₉. Then, the probability of drawing the 2 next is ¹/₉₈. So, the probability of drawing exactly those numbers in that order is ¹/₁₀₀ × ¹/₉₉ × ¹/₉₈ = ¹/_970,200.

Talk to us if that puzzles you. Now, did you automatically reach for your calculator? 

Using some strategic thinking we can use mental math to calculate that imposing product: 100 × 99 × 98. Multiplying by 10s (and 100s) is super simple since we can tack on corresponding 0s. The number 99 is the difference 100 - 1. So rewrite the product as 100 × (100 - 1) × 98. Now the associative law allows us to distribute 98 through the difference factor = 100 × (9800 - 98). Solve the difference with another strategy, since 98 = 100 - 2, the difference of 9800 - (100 - 2) is 9800 less 100 = 9700, then add back the 2 to get 9702. Finally, multiply by 100 and get 970,200 (tada!).

Did you lose your marbles or keep them? Feel free to stop by Mathnasium to talk about this and other criticial thinking and number sense approaches that we teach.

Contact:

Ruby Yao and Benedict Zoe, Mathnasium of Fort Lee
201-969-6284 (WOW-MATH), [email protected]
246 Main St. #A
Fort Lee, NJ 07024

Happily serving communities of Cliffside Park, Edgewater, Fort Lee, Leonia, Palisades Park, North Bergen, West New York, and Fairview.