We all know the feel­ing; TV sta­tions seem to be show­ing the same episodes of your favorite show over and over again. While the con­spir­a­cy-the­o­rist in me would love to believe this is true, there’s actu­al­ly a very good rea­son for this.

Remember when you were in school, and despite the appar­ent unlike­li­hood (after all, there are quite a few days in a year), two of your class­mates had their birth­days on the very same date? Actually, this isn’t unlike­ly at all; in a group of 23 or more peo­ple there is a 50% chance that at least two of their birth­days will coin­cide. For 57 or more peo­ple the chance is 99%! This is com­mon­ly known as “the birth­day para­dox”, although it’s not real­ly a para­dox at all.

The same prin­ci­ple applies to TV show episodes, and since most series have a lot less than 365 episodes, the prob­a­bil­i­ties are actu­al­ly even high­er. We’ll do the cal­cu­la­tions for a few well-known shows, but first let’s see how it works.

Let E be the set of episodes for a giv­en show. We denote the num­ber of episodes by |E| (read: “the size of E”). Now, for a giv­en num­ber of seen episodes to all be dif­fer­ent, they must be pair­wise dis­tinct. Let’s cal­cu­late the prob­a­bil­i­ty p_E(n) that n ran­dom­ly cho­sen episodes of E are pair­wise distinct.

This cal­cu­la­tion can be under­stood as the prob­a­bil­i­ty of choos­ing an unseen episode n con­sec­u­tive time. The prob­a­bil­i­ty of see­ing the same episode twice when watch­ing n episodes of E is then giv­en by 1-p_E(n). Let’s study this for a few well-known TV shows.

For Friends, which has 236 episodes, the num­ber of episodes required for a 50% chance of a repeat is 19, and watch­ing 46 episodes gives a 99% chance. For House, cur­rent­ly at 162 episodes, these num­bers are 16 and 38 episodes respec­tive­ly. For America’s longest run­ning sit­com The Simpsons, cur­rent­ly at 492 episodes, watch­ing 27 episodes gives a 50% of a repeat sneak­ing in; and watch­ing 67 episodes brings this up to 99%!

So the next time you tune in to your favorite syn­di­cat­ed TV show and are dis­ap­point­ed that you’ve already seen the episode, you can feel com­fort­ed that it’s not the net­work’s fault — it’s just math.

Extra cred­it: The birth­day para­dox can also be applied to why your iPod shuf­fle appar­ent­ly keeps choos­ing the same songs to play. Although, that choice also seems to be influ­enced by the law that any ran­dom choice with­in a playlist will pick the worst song in the list.