What Is The Probability They Are Both Red?

Stay for Lewis Carroll’s Pillow Problem at the end 🛌

The bag contains four balls each of which is colored either red or blue. So it can have all 4 red balls, all 4 blue balls or a mix of both.

If a ball is drawn at random from the bag but not replaced and then a second ball is drawn at random, the probability that both balls are blue is 1/2.

The question is, what is the probability that both balls are red?

A = 1/2, B = 1/3, C = 1/4, D = 1/6, E = 0

Read to the end for an investigation of Lewis Carroll’s Pillow Problem⭐⭐

I recommend you pause the article, grab your pen and paper, and give this a go. When you are ready, keep reading for the solution! ✒️

---

Solution

Let’s suppose b of the 4 balls are blue. There are b ways to choose a blue ball first, and this leaves b − 1 ways to choose a second blue ball.

So there are are b × (b − 1) ways in which both balls are red.

With 4 balls in the bag there are 4 × 3 = 12 ways to choose first one of the four balls and then one of the three of the remaining balls.

We are give that the probability that both balls are blue is 1/2, so we can form the following equation.

Upon expanding and factorization, we get the following.

This means either b = 3 or b = -2. Now we cannot have a negative number of blue balls, so b must be 3.

This means 3 of the balls are blue and only 1 of the balls is red, which makes it impossible to have 2 red balls. And so the probability that both balls are red is 0.

Well done!

---

Lewis Carroll’s Pillow Problems 🔴🟢

Suppose a bag contains two balls each of which is red or green with equal probability. Can you determine their colors without taking them out of the bag?

Perhaps surprisingly, we can confidently say that one ball is red and the other is green.