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Can you keep a secret? | Entry id: secret-probability |

By The Famous Brett Watson On Sat, 08 May 2004 19:45:00 +1000 |

Three may keep a secret, if two of them are dead.

— Benjamin Franklin

You may have heard that information wants to be free. Personification aside, the simple fact is that secrets are hard to keep, especially when you want to share that secret with a particular group, but not with anyone outside that group. In order for such a shared secret to remain a secret, there must be no leaks: no instances where someone in the group lets the secret slip to someone outside the group. People outside the group have generally made no promise to keep the secret, and so once the secret is out, it's likely to spread.

So far I haven't told you anything that isn't fairly obvious. In order to raise this article above an exercise in stating the obvious, I'd like to explain the secret-keeping problem in terms of probability. In order to introduce probability into the picture, we need a probabilistic model of secret-keeping. In this model, we rate our secret-keepers on a scale from zero to one, where "zero" represents someone who *never* keeps a secret, and "one" represents someone who *always* keeps a secret. Call this the "trustworthiness rating". Because secrets are divulged over time, we need to express this rating in terms of a time-frame, but that fact is not immediately important.

In order for our secret to be kept, we have to share it amongst a certain number of people, but not let the secret leak to others outside the group. If any one member leaks the secret, the secret is out; or, equivalently, all members must actively keep the secret. In terms of probability, this means that we *multiply* all the trustworthiness ratings together to find the overall trustworthiness rating of the group. The group as a whole will be as trustworthy as a single person with the same trustworthiness rating.

For example, say we have a group of twelve people, each of whom has a trustworthiness rating of 95%. As individuals, each of these people will keep a secret (for a given time) 95% of the time. That seems pretty reliable, especially if the time frame is a large one, like a year. But when you consider them as a group, their trustworthiness rating is 95% to the power of twelve (0.95^{12}), which is a mere 54%. Rationally speaking, if you wouldn't trust a person with a trustworthiness rating of 54%, you shouldn't trust a group of a dozen people, each of which has a 95% trustworthiness rating.

The trustworthiness rating is also modified with respect to time using multiplication. If a given person has a trustworthiness rating of 99% per day, then their weekly trustworthiness rating is 99% to the power of seven (0.99^{7}), which is closer to 93%. Their per-year trustworthiness rating is around 2.5%!

Finally, both the number of time-periods and number of people in the group can be multiplied together for these computations. If each member of a group has a trustworthiness rating of 99% per day, and you want a dozen such people to keep the secret for two weeks, then your chance of success is 0.99^{(12*14)}, which is about 18.5%.

Pure mathematics isn't a perfect model for secret keeping: there is a certain amount of psychology involved in persuading someone to keep a secret, for example, and this can have a significant impact on the outcome. The larger the scale of secret-keeping, however, the more relevant a mathematical model like this becomes. Like The Mathematics of Monkeys and Shakespeare, the results aren't always what you would intuitively expect them to be.