TWO QUESTIONS IN PROBABILITIES.
(
Money Puzzles)
There is perhaps no class of puzzle over which people so frequently
blunder as that which involves what is called the theory of
probabilities. I will give two simple examples of the sort of puzzle I
mean. They are really quite easy, and yet many persons are tripped up by
them. A friend recently produced five pennies and said to me: "In
throwing these five pennies at the same time, what are the chances that
at least four of the coins will turn up either all heads or all tails?"
His own solution was quite wrong, but the correct answer ought not to be
hard to discover. Another person got a wrong answer to the following
little puzzle which I heard him propound: "A man placed three sovereigns
and one shilling in a bag. How much should be paid for permission to
draw one coin from it?" It is, of course, understood that you are as
likely to draw any one of the four coins as another.
Answer:
In tossing with the five pennies all at the same time, it is obvious
that there are 32 different ways in which the coins may fall, because
the first coin may fall in either of two ways, then the second coin may
also fall in either of two ways, and so on. Therefore five 2's
multiplied together make 32. Now, how are these 32 ways made up? Here
they are:--
(a) 5 heads 1 way
(b) 5 tails 1 way
(c) 4 heads and 1 tail 5 ways
(d) 4 tails and 1 head 5 ways
(e) 3 heads and 2 tails 10 ways
(f) 3 tails and 2 heads 10 ways
Now, it will be seen that the only favourable cases are a, b, c,
and d--12 cases. The remaining 20 cases are unfavourable, because they
do not give at least four heads or four tails. Therefore the chances are
only 12 to 20 in your favour, or (which is the same thing) 3 to 5. Put
another way, you have only 3 chances out of 8.
The amount that should be paid for a draw from the bag that contains
three sovereigns and one shilling is 15s. 3d. Many persons will say
that, as one's chances of drawing a sovereign were 3 out of 4, one
should pay three-fourths of a pound, or 15s., overlooking the fact that
one must draw at least a shilling--there being no blanks.
Informational
