Pedant’s Paradox

About the series: Examining logic problems and paradoxes and dismantling them, because I am just that picky. Feel free to debate my answers. (Yes I am aware most of these have mathematical answers, but they’re dressed  in real-world examples so they can be looked at with real world logic).

The problem: If the chance of someone in the population being hit by lightning is 650,000 a year, and Kelly is hit by lightning in 2009 what are the chances she will be hit by lightning again in 2010?

The Answer: The standard answer is one in 650,000 as years do not affect each other. However this is incorrect. Kelly is no longer only part of the group “general population”, she is also part of a smaller group: people who have already been hit by lightning. The chances of that group being hit may not be the same as the general population.

This is because people who have already been hit by lightning include a subset, and in some years a majority, of people who engage in behaviours more likely to get them struck by lightning – e.g. steeplejacks, deep sea anglers, high altitude construction workers, mountaineers, and depending on your definition of lightning, electricians who immediately skew the odds.

Their chances of being struck by lightning are higher. The average (mean) chance across a group is the total number of probabilities divided by the number of members in the group. As there are more people in the group with higher risks, the average chance of being struck by lightning again is higher. By being struck by lightning, Kelly therefore enters a higher risk group.

However, even this is incorrect. What we do not know is how Kelly reacted to the first strike. If she changed jobs, decided she was not at risk, or even if she survived the strike. This is why mathematical models must be treated with caution.

The correct answer is two-part:

  1. i) given no changes in circumstance (ceteris paribus – all things being equal) her chance of being struck by lightning should be the same as the year before.
  2. ii) without knowing more about her behaviour and circumstances, that probability is impossible to determine.