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.