## Answer to November Question – Daily Lottery

If you chose 5) There is no difference, every day is equally as likely, congratulations, you have good statistical sense, and are probably quite sure of your answer, but you are nonetheless wrong – just less wrong. The correct answer is 1), tomorrow.

Counterintuitive? Here’s why.

Yes, every day is equally as likely to hit the lottery. Namely, a 1% chance.  So what is wrong?

Let’s look at the actual question.  The question was not “what day am I most likely to win the lottery?” –  it was “what day am I most likely to stop playing?”.  At first glance the two questions seem to be the same, yet there is a subtle but very important difference.

Let’s say today is Sunday. Tomorrow (Monday) I have a 1% chance to win. The day after tomorrow (Tuesday) also carries a 1% chance to win. The same goes for Wednesday, and every day after that. However, while the chance I win on Tuesday is 1%, the chance that I stop playing on Tuesday is less than that, because I must not have already won on Monday.  If I had gotten lucky on Monday and won, I wouldn’t even have had a chance to play on Tuesday, because I would have stopped already.

Each day after that, the chance that I stop on that particular day decreases accordingly, not because I’m less likely to win on that day, but because I cannot have already won any day before then. Therefore, the most likely day that I will end up stop playing is tomorrow, which carries a 1% chance. Every day after that carries a chance of less than 1%.

This is a good example of how our intuitions fail us. I stated clearly that it is not a trick question, but “you just need to understand the question”, and for good reason. The question asked was “what day am I most likely to stop playing?”, which many people immediately substituted for a much easier question, “what day am I most likely to win?”. There is a subtle but important difference, which is the hard-to-spot implied condition of previous losses. To stop playing on a day does not just mean you win that day, but also that you cannot have won before that day.

The last option “There is no difference, every day is equally as likely” is so appealing because it is a true statement.  The statement just happens to be irrelevant to the question. It is a red herring to throw you off the trail.  Similar to a mental sleight of hand, it’s a powerful technique, widely used by marketers, politicians, monthly question askers, and boyfriends/girlfriends.

## November Question: Daily Lottery

I participate in a simply daily lottery.  The rules are simple.  Every day I choose a number from 1-100, and at the end of the day, a random winning number from 1-100 is drawn.  If it matches the number I chose, I win.  The odds of winning a single game is 1 in 100, or 1%.

I was very lucky and won the lottery today. I decide to keep playing one number every day until I win again, and then stop playing for good. What day am I most likely to stop playing (by winning the lottery that day, of course)?

1. Tomorrow
2. The day after tomorrow
3. 50 days from today
4. 100 days from today
5. There is no difference, every day is equally as likely

This is not a trick question so you do not need to consider unusual scenarios – you just need to understand the question.

## Answer to October Question, Part 2

This is a great question because it forces one to think about what a moon phase really is, not just a standard answer regurgitated from a textbook. To answer this question, let’s take a look at the drawing I provided before, and mark out the surface visible from Earth, marked in red:

Next, let’s show the illuminated surfaces of the moon, marked in yellow, assuming sunlight coming from our left:

The “full moon” phase is the easiest. Only one side is illuminated by the sun, which turns out to be the visible side. You will see a bright square.

The next phase is more interesting. Two sides are illuminated now (remember, sunlight comes from our left), but we can only see one side, which is illuminated at an angle. We should still see a full square, but less bright.

At a certain point, the sun no longer illuminates the side visible to us and it goes dark. This will continue for half a lunar month cycle.

To summarize, for half a lunar month you will see no moon, and for the other half, you will see a full square moon going from dim to bright to dim.

References: