Understanding one in 100,000

What does this tiny probability look like?

The EMA stated there was a plausible link between the Oxford-AstraZeneca (Vaxzevria) vaccine and rare blood clots. The European Medicines Agency said the probability of this kind of blood clot was “very low”. With millions vaccinated, very rare side effects can emerge.

The Paul-Ehrlich-Institut reported brain blood clots with low platelets in one in 100,000 vaccinations. The MHRA figure was a little lower: at about 0.4 in 100,000. The disparity comes from coverage, case definition, study period, and population differences.

What does one in 100,000 look like?

One in 100,000 is a very small probability. It is challenging to think about this chance.

We can put 100 dots in a row. We can then stack 100 rows of those dots on top of each other. The square of 100 is ten thousand. The probability of one in 10,000 then looks like:

You should be able to see the orange marker. (Image: R Pubs)

Ten stacks of those squares is then 100,000 dots:

Spotting the orange dot may be more difficult this time. (Image: R Pubs)

What else has these odds?

There are other risks which have a similar probability:

  • Dying under general anaesthetic: This is for healthy patients having non-emergency surgery. The NHS states the chance of death is around one in 100,000 under general anaesthetic.
  • Dying in a skydiving incident: Between 2015 and 2019 in the United States, there were around 0.6 deaths per 100,000 jumps.
  • Fatal injuries at work in two years: In Great Britain, the rate of fatal injuries was about 0.5 per 100,000 workers in 2019/20.
  • Getting deep vein thrombosis from a short flight: For healthy people, the absolute risk is one in about 107,000 flights lasting less than four hours.
  • Non-fatal anaphylatic shock after the Pfizer-BioNTech vaccine: The CDC reports there were 1.1 cases per 100,000 doses in December 2020. The very rare chance of immediate issues is why people must wait after receiving vaccines.

Games of chance

Since dying is a very rare event, similar odds to one in 100,000 can be macabre. We can also think of probability-based games and chances:

  • Winning the Lotto jackpot with 450 tickets: The probability of winning the Lotto jackpot with one ticket is about one in 45,000,000. With 450 tickets in hand, like you hailed from a maths textbook, the odds of the ultimate win are around one in 100,000.
  • Guessing the last five digits of a phone number: There are five choices of ten digits. That means the chance is exactly one in 100,000.
  • Tossing a fair coin and getting 17 heads in a row: For your first attempt, the probability is around one in 131,000. If you keep going, the odds of 17 consecutive heads rise. Illusionist Derren Brown explored this concept in the programme The System.
  • Rolling six sixes and then an even number: On a fair die, there are six sides with equal chance. The probability of six sixes is one in 46,656. The chance of six sixes and an even number is one in 93,312.
  • A Royal Flush with six redrawn hands: A Royal Flush is an Ace, King, Queen, Jack, and 10 from a deck of cards. With five cards, there are 2,598,960 combinations of cards. Four of these are Royal Flushes — one in each suit. That means the probability is one in 649,740. With six redrawn hands, the probability of a Royal Flush is around one in 108,000.

One in 100,000 is a very small chance, which can be challenging to grasp.

The R code for the dot images is available on R Pubs and GitHub.

This blog looks at the use of statistics in Britain and beyond. It is written by RSS Statistical Ambassador and Chartered Statistician @anthonybmasters.

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