Here are two related
brain
teasers. The answers are given farther down the page. If these problems
are new to you, please try carefully to get the answers, then check
your answers and read the explanation. Pay attention to the feelings
that you have, or recall your feelings when you first saw questions
similar to these.
Brain Teasers:
1. You know that a family
has 2 children. You learn that the older child is a girl.
Now what is the probability that the other child is a girl?
2. You know that a family has 2 children. You learn that at
least one of them is a girl. What is the probability that
the other child is also a girl?
— Please work both problems, then scroll down for the answers —
Answer
to
Question #1

The
older child
is a girl.
The probability that the other child is a girl is 1/2.

— keep scrolling for the next
answer —
Answer
to
Question #2

At least
one
child is a girl.
Now the probability that the
other child is a girl is 1/3.

— keep scrolling —
Explanation
Both problems: if you
have no information except that there are 2 children, then there are 4
possible cases:

A

B

C

D

Older

girl

girl

boy

boy

Younger

girl

boy

girl

boy

1. If you are told that the
older child is a girl, then the possibilities are only A
and B. The probability of case A (2 girls) is one out of two, or 1/2.
2. If you are told that at
least one child is a girl, this excludes case D. The
possibilities are A, B, and C, and only in case A is the second child a
girl. This is 1 case out of 3, so the probability is 1/3.
Discussion
How did you feel the
first time that you saw these puzzles, whether today or earlier?
Just look at the 2 questions. Does the first question seem dumb? Why
does the girl's age matter? Is a big girl more likely to have a sister
than a little girl?
You are welcome to your
feelings,
whatever they are, but consider this thought. The problems may be quite
mysterious when you first encounter them. When you see the logical
answer, it works by taking the mystery out of the problem.
It may appear at first
that the
problem of the two children is a trivial amusement. The setup is simple
and you could ask the older sister what her sibling's gender
is. In reality, this is an important example for a course in
probability. For instance, it can be viewed as speaking to the notion
of independent events. If you know only that there are two children,
then the sex of the older and the sex of the younger are independent
events. If you are told that "at least one child is a girl," the gender
of the other child is no longer independent. If you learn that "the
older child is a girl," then you have definite knowledge about the
older child, but you know nothing about the younger. The younger
child's gender is again independent.
There are 3 points, then,
that
relate to color rendering:
1. Even a tiny dose of logic can create an element of surprise or
mystery.
2. When you follow through the logical answer, it takes the
mystery out of the problem. This is characteristic of analyzing
something logically. When you list the cases, the mystery of the
2child problems falls away.
3. When some of the mystery is gone out of color rendering, you may
find that it is an important subject, more important than you thought.
That's it really. Many
discussions
of color rendering convey a single fact, that the light source
"somehow" affects what you see. The inner workings of this "somehow"
are not explained; the discussion is cut short. In my presentation, I
move right into the mysterious inner workings. The way I do this is to
bring many tidbits of fact into the discussion. More facts offer more
chances for something to seem temporarily mysterious.
>>
proceed to another similar puzzle.
