Probability An Event Will Occur after N TimesI’ll be the first to admit that I’m a geek – a Ph.D. in physics was not an idle pastime. But there’s one area of study where I really revel in geekiness – statistics. (Go ahead and roll your eyes.) WARNING: There will be equations.

There are so many applications of statistics to everyday life that the list would be endless – and most people would be surprised at the results because so many times statistics seems to be counter –intuitive. I realized this just two days ago when I received a comment on my post about General Mills and Gluten-Free Cheerios – Statistics That Really Matter.  I stated in the post that if a person ate one box of Cheerios per month for a year, they would have a 20-50% chance of eating gluten. Ari wrote to say the chance was 2-5%, not 20-50% in my example. Ari didn’t really understand, because the 2-5% was for a SINGLE box – so how did I get to 20-50%? Let me start with a story.

In World War II, American air combat crews were concerned that the attitude for rotation out of combat was, as summed up by Lt. Gen Millard Harmon,

“We give them no reasonable assurance as to how long they will have to carry the ball. To them there appears no end – just on and on till the Jap gets them.”

In Europe, an attempt was made to give the heavy bomber crews hope by telling them that they had only a 1% chance of being shot down on any given mission – not really true since some missions had rates greater than 10%. Then someone calculated that a 1% chance on each mission meant that the odds of going home were 50% given the number of missions the crews had to fly at that time.

In both World War II and with Gluten-free Cheerios, the calculation is the same. I’ll illustrate.

Let X be the chance of being shot down or getting a box of Gluten-free Cheerios that contains more than the FDA limit of gluten. Then

X = Chance of happening once
1 – X = Chance of not happening once

What’s the probability the second time? It’s

X = Chance of happening once
1 – X = Chance of not happening once

Yes, exactly the same as the first time. So we have four possible outcomes if we take two events, whether flying a B-17 or eating Cheerios:

  1. It will happen the first time and the second time ( X * X )
  2. It will happen the first time but not the second time ( X * (1 – X) )
  3. It will not happen the first time and will happen the second time ( ( 1 – X ) * X )
  4. It will not happen the first time or the second time. ( 1 – X ) * ( 1 – X )

SimTable of Probabilitiesply stated, the probability of the first event times the second event.

In general, the probability that the event won’t happen in N times is ( 1 – X ) to the Nth power. This is simple to do in Excel. The results for up to 40 times for a 1%, 2%, 5% and 10% chance of something happening a single time are shown here in table form and in graph form above.

As you can see, for a 2% chance a single time, after 12 times there’s a 22% chance that the event will have happened. For 5%, it’s nearly 50%.

If you are a celiac, those are the odds you will be glutened if you are a regular consumer of Gluten-free Cheerios. If you were a B-17 crew member over Europe, those were the odds you would be shot down before you could go home.

Of course, you might be lucky – if you play the odds.

A physicist by trade, author by choice, a born teacher, a retired veteran, and an adamant problem solver, Frank has helped the White House, federal agencies, military offices, historical museums, manufacturers, and over 250 technology startups get stuff done, communicate effectively, and find practical solutions that work for them. In his spare time, he makes sawdust and watches Godzilla movies.