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Two people, person "A" and person "B" each have
two coins. They each flip both of their two coins.
When person "A" was asked
if at least one of the two coins landed "heads up", the reply
was "Yes; in fact the first coin I flipped landed heads up."
When person "B" was asked the same question, the reply
was a simple "Yes".
What is the chance that person "A" has both coins heads up? According to the "Genius" Marilyn Vos Savant, assuming at least one of the coins landed heads up for each person, the correct answer is: Person "A" has a 50% chance of having two heads,
Person "B" has a 33% chance of having two heads. Roger Amidon says the correct answer is:
Assuming at least one of the coins landed heads up for each person,
both person "A" and person "B" have a 50% chance
of having both coins land heads up.
Roger's Proof:
The chance that either person "A" or person "B" has both
coins "heads up" is the same. Here's why:
When person "A" says the first coin flipped came up heads, it does not
change the "other" coin's chance of being heads or tails.
Person "B" said that one of their two coins also came up heads. We
don't know if it was the first coin or the second. Person "B" likes
to keep secrets perhaps. The chance that the "other"
coin is also heads is still the same for either person "A"
or "B". Just because person "A" volunteered
some additional information, it doesn't affect how the "other"
coin landed - for either person.
In both cases, we have a "known" coin (heads) and an "unknown"
coin (head or tails). The fact that person "A" told us the first coin flipped
is the "known A" coin has no affect on the "unknown A" coin.
It still can be heads or tails. Person "B" also has a "known" coin
(heads) and an "unknown" coin. The fact that we don't know if it was the first
or the second coin "B" flipped does not alter the condition of the
"unknown B" coin. This "unknown" coin - for
both person "A" and person "B" - still has
it's own 50-50 chance of being heads up.
Both person "A" and person "B" have a 50% chance of having
both coins land heads up.
(In Marilyn's proof, "TH" and "HT" are really the same variable. She counts it twice.)
Marilyn's proof:
Person "A" can have 1 of the following possible 2 combinations of coin flips:
Therefore, chances of having two "heads" are 1 out of 2 for person "A", and 1 out of 3 for person "B".
Is this logically correct? Do you agree with Roger or Marilyn? (Please vote, because with some 200 votes so far, we are about even!) |