• First, write down the outcome of a series of 100 imaginary coin flips. H for heads and T for tails. Be sure to come up with the imaginary coin flips yourself! When you've finished your 100 'coin flips' and have a series of 100 letters, read the story below.

While walking in the woods, a statistician named Goldilocks wanders into a cottage and discovers three bears. The bears, being hungry, threaten to eat the young lady, but Goldilocks begs them to give her a chance to win her freedom.

The bears agree. While Mama Bear and Papa Bear block Goldilocks' view, Baby Bear tosses a coin 30 times and records the results. He then makes up two other (fake) sequences of heads and tails, and gives Goldilocks a piece of paper that shows all three sequences. Papa Bear growls, "If you can determine which sequence came from the real coin toss, we will let you go. Otherwise we will eat you for dinner, for I have grown tired of porridge."

Here are the three sequences of heads (H) and tails (T) that the bears present to Goldilocks. Each of the sequences contain 16 heads and 14 tails.

H H H H H H H H H H H H H H H H T T T T T T T T T T T T T T

H T H T H T H T H T H T H T H T H T H T H T H T H T H T H H

H T T H H H T T T T T T T H H H T H T H H H T H H H T H T H

Goldilocks studies the three sequences and tells Papa Bear:

"The first sequence is "too hot." It contains 16 heads followed by 14 tails. I would not expect such long sequences of heads and tails. Similarly, the second sequence is "too cold." It alternates between heads and tails like clockwork. The third sequence is "just right." It matches my intuitive notion of a random sequence of two categories: many short subsequences interlaced with a few longer subsequences. I think that the third sequence is real."

She had chosen correctly. The three bears, impressed by her statistical knowledge, set Goldilocks free and—once again—reluctantly ate porridge for dinner.

. . .

You can quantify Goldilocks' intuitive notions by defining a run as a sequence of consecutive trials that result in the same value. The first sequence has two runs: a run of heads followed by a run of tails. The second sequence has 29 runs. The third sequence has 15 runs: eight runs of heads and seven runs of tails.

It turns out that you can calculate the expected number of runs in a random sequence that has n heads and m tails. The expected number of runs is E(R) = 2nm / ((n+m) + 1). The three sequences have n = 16 heads and m = 14 tails, so the expected number of runs is 15.9. So Goldilocks' intuition was correct: the first sequence does not have enough runs, whereas the second has too many. The third sequence has 15 runs, which is close to the expected value.

(Original story available at: https://blogs.sas.com/content/iml/2013/10/09/how-to-tell-if-a-sequence-is-random.html)

• Now look at your series of imaginary coin flips. How random is it really? Use the equation below to find out.

• random.org (spend some time exploring different situations of randomness on this site)