Which scenario is best to achieve objective?

Question

Objective: To witness lightning striking a 1 meter area of the ocean with the least amount of wait time.

Scenario 1, a 1 in 10k chance with lightning occurring once every quarter.

Scenario 2, a 1 in 200k chance with lightning occuring every day.

Answer 1

Hey, Stan!

Questions like these are basically the same thing as questions that ask "If you flip a coin, what are the chances you get heads x times in a row?" This one is just written the other way around.

Instead, this one asks, "If you watch a patch of ocean, how long are you likely to go without seeing lightning?" Or in other words, "How long can you flip a coin and only get tails?"

Here, we're looking at two different coins: one of them has 9,999 tails, and the other one has 199,999 tails.

Let's call the first one Coin A. We get to flip it four times per year, or roughly every 90 days.

Coin B, we get to flip every day.

Looks to me like we'll know the answer after 90 days. We have to figure out whether, even though it has a lower individual probability per flip of coming up heads, things might look different after we flip Coin B 90 times in a row.

Let's find out!

The chance of lightning not striking in the first case (the Coin A case) is 9,999/10,000 after 90 days. Therefore, the chances of lightning striking in the first case is 1 - (9,999/10,000) = 1/10,000. Simple enough.

It's a little more complicated when you string individual probabilities together one after the other. In Case B, the chance for lightning to not strike on Day 1 is 199,999/200,000. Same thing on Day 2. Or on Day 3. That's 0.999995 on any given day.

The joint probability for lightning to strike on neither Day 1 nor Day 2 is (199,999/200,000) * (199,999/200,000), which my calculator says is 0.999990000025. Not much of a change from 0.999995, but it's a start.

The chance for lightning to strike on neither Day 1 nor Day 2 nor Day 3 is (199,999/200,000) * (199,999/200,000) * (199,999/200,000), and you can see where we're headed from here. Every successive day is another successive multiplication by 199,999/200,000.

For short, the formula for calculating the chances of lightning never striking that little patch of ocean for ninety days straight is (199,999/200,000)^90. If you want to plug it into Excel, use

= (199,999/200,000)^90

Excel tells me the answer is 0.995501 blah blah something something.

So if 0.995501 etc. is the chance that lightning won't strike in 90 days in Case B, then (1-.995501 etc.) is the chance that lightning will strike in ninety days in Case B. Excel tells me that's equal to .00044989 etc., or about .045%. Not quite half a percent.

Not by any means likely. A chance somewhere between fat and slim. But keep in mind that the probability of lightning striking on any single one of those days was only 0.0005%. The simple fact of being able to try and try again 90 times raised the probability that lightning would strike by a factor of almost 9,000.

Now let's see which case has the greater chance of seeing a lightning strike!

Case A is 1/1000. One in 10,000 in decimal notation is .0001, or .01%

Case B is .045%

Since the chances of having seen lightning over 90 days are four times higher in Case B than in Case A, it looks like it'll probably take less time to see lightning in Case B!

Whaddaya know?

Protip #1: Running scads of trials (like flipping the coin a lot of times) is almost always the key to finding at least one quirky success. It rarely matters how bad the odds are against you if you only have to succeed once (like getting heads, or watching lightning strike), provided you get a lot of chances to succeed.

Protip #2: Questions like these are set up to frustrate you into getting caught up in how you're supposed to calculate the joint probabilities of success. Forget that. You solve them by calculating the joint probability of failure and then subtracting that from 1.

Protip #3: Sometimes questions like these are even more complex. They may ask, "What are the chances of lightning striking twice over 90 days" or something even fancier. There are formulas for it in your textbook which are generalizations of the one we used up above. You'll do well to practice and memorize them. Then every problem like this will become a snap!

P.S. If you really want to get geeky in Excel, there's an optimization function you can play with that will tell you how many days it will take for the chances of Case B to equal the chances of Case A. In your copious free time, of course.

Good luck!