80 of 100: Basketball Fallacy

$P(option 1) = A+B$

$P(option 2) = B$

$A+B > B$

$P(option 1) > P(option 2)$

If he is the former, he is not necessarily the latter.

If he is the latter, he is necessarily the former.

The former must be at least as likely as the latter.

P(A and B) <= P(A) (Probabilities being <= 1)

Since there is about 7.1 billion people in the world, we could just divide 7.1 billion by 300,000 to get 23,600 7 footers(a statistics and worldwide assumption). but very very few of them are basketball players.

so, John is 7 feet--this has the most probability.

$\textbf{Proof using words}$

Only a small fraction of people are over 7 feet tall. And, among these 'over-7-feet' people, only a small fraction are professional basketball players. So, people satisfying both the conditions are less likely to be found.

$\textbf{Proof using maths}$

Let, E be the event of people being more than 7 feet tall and F be the event of being a pro basketball player. We can use conditional probability now.

$P(E \cap F)=P(E)P(F|E) \leq P(E)$ $\implies P(E \cap F) \leq P(E)$

It's because $0 \leq P(F|E) \leq 1$ .

[ $P(F|E)$ is the probability of a person being a basketballer given that he's more than 7 feet tall.]

In order for the second option to occur, two stipends must be met, whereas in the first, only one of these must be true. Therefore, it must be more likely that the first option occurs.

John does not have to be a professional basketball player simply because he is holding a basketball.

The answer is simple, the less assumption we make about a situiation, the more likely it is true.

If answer #2 is true, then so is answer #1. Therefore, #1 is more likely than #2.

So A>B

The first statement is true in 1/2 chance while the second statement is only true in 1/4 chance...

Option B has two assumptions that have to be true in order for it to be the correct answer, while Option A only has one. Statistically, one assumption is more likely to be true than two. Additionally, in order for Option B to be true, Option A must also be true. So if the odds of being 7 feet tall are 1/2, and the odds of someone being a professional basketball player is 1/2, Option A has a 1/2 chance of being true, while Option B has only a 1/4 chance. If we factor in the statement that if Option B is correct then Option A is as well, this boosts Option A's chance to 3/4, with Option B's odds at only 1/4.

John is seen shooting a basketball.

There are no other people or objects (Count out the clothes, the shoes, himself, and the basketball) to judge his height with.

And we cannot prove that he is a professional basketball player with just this picture.

So to make the least assumptions, choose the more general choice, which is " John is over seven feet tall "

Not all 7feet or taller people are basketball players so 7feet or taller people have a higher probability than the intersection of athletes and 7 feet or taller people

Being >7ft AND a professional basketball player is a subset of the set of being an individual that is >7ft. To be >7ft is already a low likelihood, but to have the height characteristic + the skill/talent to play professional basketball is an even greater anomaly. Furthermore, think about the average NBA height: it's certainly nowhere near 7ft and therefore even if one makes it to the professional level, on average, they're <7ft.

The truth is usually the easier to understand answer. Also more probable.

It's easy to be tempted to think that the relative proportion of 7-foot people and basketball players in the general population is important, or that the tendency of basketball players to be very tall is important, but it really is not.

The crux of it is: The statement "John is a professional basketball player and is over 7 feet tall" can only be true if the statement "John is over 7 feet tall" is also true. There is no possible world in which John is a professional basketball player and is over 7 feet tall, and yet somehow John is not over 7 feet tall. And if every case where A is true is also a case where B is true, A cannot be more likely than B.

This leaves the option that the two statements are equally likely. That would be the case in a world where everyone who is over 7 feet tall is a professional basketball player. But we do not live in such a world.

I just assumed that if he was the latter, he would be grabbing the ball with both hands :^) ... also, there is nothing that proves he's a pro player

The second statement requires something extra to be true, so at most the second statement is equally likely (and then only if all 7-footers are also professional basketball players, which is definitely false as a matter of fact).

So, we can conclude the 1st statement is more likely to be true.

Occam's Razor tells us that we should believe the possibility with fewer assumptions. the former has 1 and the latter has two so the answer should be the former

1st sentence has only one condition that must be fullfilled in order to be true. 2nd, although, has two conditions. It is quite obvious that 1st sentence is more likely to be true than the 2nd one. I don't see any other approach to this problem. What about you?

At first my reasoning was that there are less current NBA players in the world (i.e. about 400) than there are current 7-footers (about 4,000 but who knows really). After re-reading the choices, I realized that the second choice said professional basketball players . There are probably, at least, tens of thousands of people in the world that get paid to play basketball. Ugh... so then I consulted my magic 8-ball: question-"Is the first choice correct?", answer-"Concentrate and ask again." Thanks magic 8-ball ...

After re-reading the choices again, I thought the misleading temptation was to assume that given the fact that John was a paid professional basketball player, what was the probability that he would be over 7-foot tall. However, the second choice doesn't actually say that. I finally interpreted the second choice to say that John was a member of the subset of 7-footers in the world that get paid to play basketball. Logically, that has to be fewer people than the set of people in the world that are 7-feet tall. So I guessed the first choice: John is over 7-foot tall , and consulted the magic 8-ball (while holding a lucky charm this time) ... "Signs point to yes".

Yay! I'm glad there wasn't a third distractor, something like "John is under 7-foot tall AND marries a woman who is over 7-foot tall". Yikes!!!

You can assume that he is 7 foot tall because both answers have the statement, but just because he has a basketball in his hands doesn't show us that he's a basketball player.

John is 7 feet tall = P(S) John Play Basketball = P(B) John is 7 feet tall and plays basketball= P ( S ∩ B )

Of course, P(S) > P ( S ∩ B ) . So, it is more likely that John is 7 feet tall.

The second statement contains 2 conditions, while the first one contains only 1 of those 2 conditions.

It is more likely that 1 condition holds true, than that both do.

John can be 7 feet tall because there are such people enough in the world. So probability is higher

In other case there are few basketball players. So probability is less.

There are very few professional basketball players

But there are many tall people