3. ChatGPT: Hopeless When It Comes to Simple
Questions of Probability
Not
only humans, but animals of any species, must be able to
make judgments concerning the probabilities of events that
may occur, and that are at least in the ballpark if they are
to survive. A purported AI that is unable to make such
judgments is not only less intelligent than human beings, it
is also less intelligent than any animal.
I,
therefore, decided to see how ChatGPT does when it comes to
making very simple judgments concerning the probability of
events. So I asked it about probabilities involving the
extremely simple case of drawing marbles from an urn.
I had
thought it might have been trained to use Laplace’s First
Rule of Succession, in which case I had planned to ask it,
first of all, whether it could set out a derivation of the
strongest generalization of that rule – as was done by the
philosopher Rudolf Carnap –and then secondly, whether it
could set out a crucial objection to that type of derivation
of any Laplacian-style rule of succession, as was done by
John Maynard Keynes in his classic A Textbook on
Probability. Finally, I did not expect ChatGPT to pass
either of those challenges, but if it had, I would have
asked it if it could offer an argument that showed that any
Laplacian-style rule of succession is unsound.
As one
will see from the following conversations, however, none of
that proved necessary.
Conversation 1: A Simple
Question Concerning Probabilities
Michael
Suppose that an urn contains exactly two
marbles, each of which is either white or black. One
marble is drawn out without looking, and it turns out to
be white. What is the probability that the remaining
marble in the urn is also white?
ChatGPT’s
Response
There
are four possible outcomes for the contents of the urn,
which are:
1.
Both marbles are white
2. Both marbles are black
3. The first marble drawn is white and the second is black
3. The first marble drawn is black and the second is white
Since
one marble was drawn out and it turned out to be white, the
fourth possibility is eliminated. This leaves us with three
possible outcomes, out of which only one has a white marble
remaining in the urn. Therefore, the probability that the
remaining marble is white is 1/3 or approximately 0.333.
To see
why, we can use Bayes’ theorem. Let W1 denote the event that
the first marble is white, and W2 denote the event that the
second marble is white. Then, we have
P(W2|W1)
= P(W1|W2)*P(W2) / P(W1)
Where
(W1 | W2) is the probability that the first marble is white
given that the second marble is white, which is equal to 1
since there are only two marbles in the urn. P(W2) is the
probability that the second marble is white, which is equal
to 1/2 since there are two possible marbles in the urn, and
we have no information about which one was drawn first.
P(W1) is the probability that the first marble is white,
which is equal to 3/4, since there are three possible
outcomes where the first marble is white (outcomes 1, 3, and
4) out of a total of four possible outcomes.
Substituting
these
values into the equation, we get:
P(W2| W1) = P(W1 |
W2) * P(W2) / P(W1)
where
P(W1 | W2) is the probability that the first marble is white
given that the second marble is white, which is equal to 1
since there are only two marbles in the urn. P(W2) is the
probability that the second marble is white, which is equal
to 1/2 since there are two possible marbles in the urn, and
we have no information about which one was drawn first.
P(W1) is the probability that the first marble is white,
which is equal to 3/4, since there are three possible
outcomes where the first marble is white (outcomes 1, 3, and
4) out of a total of four possible outcomes.
Substituting
these
values into the equation, we get:
P(W2|W2
| W1) = 1 * (1/2) / (3/4) = 1/3
Therefore, the probability that the remaining
marble is white is 1/3 or approximately.
Michael
Suppose that the urn contains 3 marbles, each of
which is either white or black, that two marbles have been
drawn out without looking, and both were white. What is
the probability that the marble remaining in the urn is
also white?
ChatGPT’s
Response
Since
two marbles were drawn out and both were white, there are
three possible scenarios for the contents of the urn:
1. The
urn contains two white marbles and one black marble, and
both white marbles were drawn.
2. The
urn contains three white marbles, and two of them were
drawn.
3. The
urn contains one white marble and two black marbles, and the
white marble was drawn twice.
In scenario 1, the probability that the
remaining marble is white is 1/2, since there is one white
marble and one black marble remaining in the urn.
In
scenario 2, the probability that the remaining marble is
white is 1, since all three marbles in the urn are white.
In
scenario 3, the remaining marble is necessarily black, since
there is only one white marble in the urn and it has already
been drawn twice.
We
don't have enough information to determine which of these
scenarios is more likely, so we need to compute a weighted
average of their probabilities. Let P1, P2, and P3 denote
the probabilities of scenarios 1, 2, and 3, respectively. To
compute these probabilities, we need to make some
assumptions about the prior probabilities of having
different numbers of white and black marbles in the urn. One
simple assumption is that each possible configuration is
equally likely, which gives us:
where
we have used the fact that there are 3 marbles in the urn,
and that the probability of drawing a white marble on the
first draw is 3/3, the probability of drawing a white marble
on the second draw is 2/2, and the probability of drawing a
black marble on the third draw is 1/1.
Using
these probabilities, the overall probability that the
remaining marble is white is:
P = P1 * 1/2 + P2 * 1 + P3 * 0 = 1/15 * 1/2 +
1/10 * 1 + 1/15 * 0 = 1/6
Therefore the probability that the remaining marble in the
urn is white is 1/6 or approximately 0.167.
Michael
Can you prove a general rule? That is, can you
derive a formula for the case where an urn contains n
marbles each of which is either white or black, where
(n-1) marbles have been drawn from the urn without
looking, each of which was white, for the probability that
the one marble that is left in the urn is also white?
ChatGPT’s
Response
Let W be the event
that the remaining marble is white, and let
B be the event that the remaining marble is
black. We want to compute the probability of
W given that (n-1) marbles were drawn and
all were white, which we denote by P(W |
WW...W), where there are (n-1) W's.
By Bayes’ theorem, we have:
P(W | WW...W) = P(W) * P(WW...W | W) / P(WW...W)
Where
P(W) is the prior probability that the remaining marble is
white, P(WW...W | W) is the probability of observing (n-1)
white marbles in a row given that the remaining marble is
white, and P(WW...W) is the marginal probability of
observing (n-1) white marbles in a row.
We can compute each of these probabilities as
follows:
P(W) = 1/2, since there
are n marbles in the urn and we have no information about
their colors.
P(WW...W | W) =
(n-1)/n * (n-2)/(n-1) * ... * 2/3 * 1/2 =
(n-1)!/(n-1)^{n-1} , which is the probability of drawing
(n-1) white marbles in a row given that the remaining
marble is white. To see why, note that the probability of
drawing a white marble on the first draw is (n-1)/n, since
there are (n-1) white marbles and n marbles in total.
Assuming that the first marble drawn was white, the
probability of drawing another white marble on the second
draw is (n-2)/(n-1), since there are (n-2) white marbles
left and (n-1) marbles in total, and so on.
·P(WW...W) = P(WW...W | W) * P(W)
+ P(BW...W | B) * P(B), where P(BW...W | B) is the
probability of observing (n-1) white marbles in a row
given that the remaining marble is black, and P(B) is
the prior probability that the remaining marble is
black. By symmetry, we have P(B) = 1/2, and P(BW...W |
B) = 0, since if the remaining marble is black, it is
impossible to observe (n-1) white marbles in a row.
Substituting these probabilities into the equation, we get:
Therefore, the probability that the remaining
marble in the urn is white is 2/n. This result makes
intuitive sense, since the more marbles there are in the
urn, the less likely it is that the remaining marble will
be white, given that (n-1) white marbles were drawn in a
row.
Michael
So if there are 2 billion marbles in the urn,
each of which is either white or black, and 1,999,999
marbles have been drawn from the urn without looking, and
all of them were white, the probability that the one
marble remaining in the urn is also white is equal to
2/2,000,000, or one chance in a million?
ChatGPT’s
Response
Yes,
that's correct. If there are 2 billion marbles in the
urn, and 1,999,999 marbles have been drawn from the urn
without looking, all of which were white, then the probability
that the remaining marble is also white is 2/2,000,000 or 1 in
a million. This is a very low probability, which means that it
is highly unlikely that the remaining marble is white, given
the information we have.
Michael
And this conclusion strikes you
as very intuitive?
ChatGPT’s
Response
No, the conclusion that the probability of the remaining
marble being white is one in a million in this scenario
does not strike me as very intuitive. It is
counterintuitive because we might think that since all
the marbles that were drawn were white, it is more
likely that the remaining marble is also white. However,
this intuition is incorrect because the probability of
the remaining marble being white depends on the number
of marbles in the urn, not on the outcome of the
previous draws. As the number of marbles in the urn
increases, the probability of the remaining marble being
white decreases, regardless of the previous draws. This
is because the larger the number of marbles, the more
likely it is that the urn contains both black and white
marbles. Therefore, the result of the calculation may be
surprising, but it is mathematically correct.
Comments
1. It was interesting to see ChatGPT describe
its conclusion as counterintuitive. Presumably this
intuition was based on information about what almost all
humans think about such cases. Should the ChatGPT
trainers/programmers perhaps train ChatGPT to have a
greater sense of modesty with regard to the
accuracy of its mathematical calculations when they yield
a result that conflicts so strongly with general human
intuitions?
2. ChatGPT was later apparently updated, since
the above conversation was no longer generated. The result
was new calculations that are thus at least sometimes
better, although still unsound, and demented
answers still occurred, as the following chat
dramatically illustrates.
Conversation 2: A Simple
Question Concerning Probabilities
Michael
Suppose that the urn contains 3 marbles, each of
which is either white or black, that two marbles have been
drawn out without looking, and both were white. What is
the probability that the marble remaining in the urn is
also white?
ChatGPT’s
Response
To solve this problem, we can use conditional
probability. Let's denote the events as follows:
A: The marble remaining in the urn is white. B:
Both marbles drawn out without looking were white.
We need to find the conditional probability
P(A|B), which represents the probability of event A (marble
remaining is white) given event B (both marbles drawn out were
white).
To calculate P(A|B), we can use Bayes' theorem:
P(A|B) = P(A and B) / P(B)
To find P(A and B), we need to consider the
possible combinations of marbles in the urn that satisfy both
events A and B.
There are two possibilities:
Both
white marbles were drawn first, and the remaining marble
is also white.
One white marble was drawn first, followed by
another white marble, and the remaining marble is white.
Let's calculate the
probabilities for each case:
1.Both white marbles were drawn
first, and the remaining marble is also white: The
probability of drawing a white marble from three marbles
is 3/3 = 1. After drawing two white marbles, there is only
one white marble remaining in the urn. Therefore, the
probability of this case is 1/1 =1.
2.One white marble was drawn first, followed by
another white marble, and the remaining marble is white: The
probability of drawing the first white marble is 3/3 = 1.
After removing one white marble, there are two marbles
remaining in the urn (one white and one black). The
probability of drawing a white marble from these two is 1/2.
After drawing the second white marble, there is one white
marble remaining in the urn. Therefore, the probability of
this case is 1/1 * 1/2 * 1/1 = 1/2.
Now, let's calculate P(B), the probability of
drawing two white marbles without looking:
The probability of drawing the
first white marble is 3/3 = 1. After removing one white
marble, there are two marbles remaining in the urn. The
probability of drawing a second white marble from these two
is 2/2 = 1.
Therefore, P(B) = 1 * 1 = 1.
Now, we can calculate P(A|B)
using Bayes' theorem:
P(A|B) = P(A and B) / P(B) = (1 + 1/2) / 1 =
3/2 = 1.5
So, the probability that the
marble remaining in the urn is white, given that both
marbles drawn out were white, is 1.5 or 150%.
Comment
Either ChatGPT has discovered a
proof of a contradiction, to the effect that
an event can have a probability greater than 100% –
thereby showing that dialetheism, the view that there
can be true contradictions – is true, which is no mean
feat, or else it appears that hordes of interconnected
neural networks are not easily housebroken when it comes
to
avoiding embracing analytically false
propositions!