Since I have been taking about probabilities recently, here is a little puzzle for people to ponder overnight.
Suppose you live in a place that has a constant chance of being struck by lightning at any time through the year. Suppose that the strikes are random: every day the chance of a strike is the same, and the rate works out to one strike a month. Your house is hit by lightning today, Monday. What is the most likely day for the next bolt to strike your house?
Feel free to post your answers in the comments. I will give the answer tomorrow along with a discussion.
Blogs with homework. Now there’s a novel concept!
Update: The solution and discussion are posted here.
After you rebuild.
Unless you properly install a lightning rod the next time.
if the average is one strike a month then there’s already a triply-redundant lightning rod system, or the frame’d never gone up without someone getting killed.
i never got past statistics 101 and can’t be bothered to google it up now, though i really should. i’m going to say there is no such thing as a “most likely day” for the next lightning strike, per the problem definition: constant chance of a strike throughout the year.
Ah, a brain teaser about the Gambler’s Fallacy. You ask, “What is the most likely day for the next bolt to strike your house?” Well, since you also say that”every day the chance of a strike is the same,” then no day is more likely than any other.
I propose that the next strike is most likely to happen tomorrow (Tuesday). Any further out, the chance of ANY strike may be constant, but that it will be the NEXT strike will be reduced by the chance of another strike happening between them.
The most likely day is Monday (as in, today). Reasoning: you already know there’s a lightning storm happening today. You can’t say the same for any other day of the month. 😀
I agree. Put explicitly:
If the probability is the same every day (1/30th = 0.033333…), then the probability of the next bolt being on Tuesday is 0.033333. The probability of it being on wednesday is 1/30th minus the fraction of THOSE events which also had one the day before (1/30*1/30 = 0.001111) so 0.033333 -- 0.001111 = 0.0322222. Using induction the probability will continue to decay. You can ignore the compounding effect of following weeks because Tuesday will always be larger than the following day, so Wed-Mon will always lose.
I think Monday as well, but for a different reason.
A “single” bolt is already several strikes. The next strike was probably milliseconds after the first 😀
I think the answer depends on what time on Monday you ask the question. If it’s before approximately 12:01:33am then I’d say the most likely day would be Monday again. On or after 12:01:33am then in purely probabilistic terms Tuesday seems the most likely day.
Calculations: chance of being hit by lightning on a particular day has been given as 12/365.25 = 0.032854. Chance of not being hit that day is therefore 1 -- 0.032854 = 0.967146. Chance of a lightning strike not occurring until the next day is then 0.032854 (chance of a strike) * 0.967146 (chance of no strike on the first day) = 0.031775. Assuming every time of day is equally likely for a lightning strike, if we ask the question earlier than (0.032854 -- 0.031775 = 0.001079) of a day (which comes to about 1 minute and 33 seconds) then the chance is highest that we’re going to get another lightning strike the same day we ask the question (Monday).
Having said that, the original problem didn’t specify when on Monday we asked the question, so assuming it’s all equally likely then on average we’re going to be halfway through Monday when the question is asked, so the most likely day for the next strike is still Tuesday, as John and Jared both said.
Of course none of this takes into account that if your house has just been struck by lightning then the conditions are ripe for a second strike!
I was going to say Tuesday, if I understand the problem correctly to be saying what day of the week is most likely.
While Karl’s analysis isn’t wrong, as is established this applies only if the strike occurs ridiculously early on that monday.
Since only a small percentage of Monday occurs before that cut-off, it is unlikely to be before that cut-off. Thus, the likeliest outcome is that Tuesday is going to be the day of the week of the next strike.
How bout that?
The gambler’s fallacy immediately came to mind when I read this, too.
If every day the chance of a strike is the same, and the rate works out to one strike a month, then I honestly don’t see any reason why there can’t be an equally likely chance of a strike still happening on any other day.
Probability doesn’t preclude that you get zero strikes in a month, either. It also doesn’t dictate that you can’t get 30 strikes in a month. An average is just that: an average. There will be values bigger than the average and lower than the average.
Looking forward to Mano’s follow-up on this!
Given:
-- There is a constant chance of being struck by lightning at any time through the year.
-- The lightning strikes are random.
-- Every day the chance of a strike is the same.
-- The house is hit by lightning today, Monday.
Assumption(s):
-- Today’s, i.e., Monday’s lighting strike is the only lighting strike of the day.
Find:
What is the most likely day for the next bolt to strike your house?
Solution:
-- The days of the occurrence of the event are mutually exclusive.
-- The day that the second event occurs is memoryless of the first, i.e., today, Monday.
-- Therefore, each day of the week has an equal, i.e., 1/7 %, chance for the next bolt to strike my house.
I agree with Madison. The lightning has no memory. Any day is as likely as any other day.
If each day you have a chance of a strike, it will be less likely to go 5 days WITHOUT a strike than 4 (which again is less likely then 3 etc). Therefore the most likely next strike is Tuesday, assuming another strike on Monday doesn’t count / cannot happen.
It sort of boils down to consecutive coin flips.
Let’s put the probability of a lightning strike on any given day at 1/30th. Then the probability of a strikeless day is 29/30th.
The probability that the next strike occurs on Tuesday is 1/30th which is 0.033333.
The probability that the next strike occurs on Wednesday is 29/30th (Tuesday must be a strikeless day) times 1/30th which is 0.032222.
The probability that the next strike occurs on Thursday is 29/30th times 29/30th times 1/30th which is 0.31148.
Etc.
The most likely day is therefore Tuesday. Indicidentally, this means that if you haven’t had a lightning strike on the current day, and the day is not over yet, you have a higher probability of a lightning strike happening that day (still 1/30th) than you have for the next day (29/30th for today times 1/30th for tomorrow).
My house was build from wood and it completely burned down the first time it was struck. Moreover that place sucks to live. I will move to a place with much better.
Conclusion: my house will never be struck again.
must be: with much better weather.
Except for that word “next”.
By your reasoning, the NEXT strike is just as likely to occur in a million years time as the following day.
That is clearly false.
The answer is either Monday or Tuesday, depending on when the strike occurred on Monday. Clearly, if the strike on Monday ocurred just before midnight, the answer would be Tuesday.
I think the answer is meant to be Tuesday and to avoid confusion, the puzzle statement should have included the line “there is never more than one strike per day”.
I just asked my wife this puzzle and, without hesitation, she answered “Tuesday”.
(I added the line “there is never more than one strike in a day”)
(I also omitted that the chance of a strike is one per month -- this is a useless piece of information unless it is intended to confuse)
By the way, if you calculate the probabilities for all the days up to infinity (perhaps not that far) and then add those probabilities, you end up with 1.00000. Meaning that eventually you will get a lightning strike, but you cannot be certain when.
The “strange” part is, of course, that regardless of when the last lightning strike was, the probability of lightning striking the next day will always be 1/30th.
“Indicidentally, this means that if you haven’t had a lightning strike on the current day, and the day is not over yet, you have a higher probability of a lightning strike happening that day than you have for the next day ”
That is clearly not correct.
I you haven’t had a strike yet on Monday and it’s a second before midnight, do you still think you’re more likely to have a strike on Monday than on Tuesday?
Pinker:
“I gave the question to a hundred Internet users, with the word next italicized so they couldn’t miss it. Sixty-seven picked the option “every day has the same chance.” But that answer, though intuitively compelling, is wrong. If every day were equally likely to be the next one, then a day a thousand years from now would be just as likely as a day a month from now.”
BillyJoe:
Except for that word “next”.
By your reasoning, the NEXT strike is just as likely to occur in a million years time as the following day.
That is clearly false.
And I swear I haven’t read Pinker’s book!
Besides, doesn’t it all depend on when the next act of sodomy occurs in (or near) the house?
Yes. The initial problem was stated as “every day the chance of a strike is the same”. It uses the artificial construct of “day” meaning 00:00:00 through 23:59:59. If that had read “within any period of 24 consecutive hours” then you would have been absolutely correct.
The “day” construct is artificial because the initial problem doesn’t state a time for the lightning strike on Monday. It uses a day as a non-divisible time unit. So within the definition of this problem my statement is correct.