A bag contains \(N\) sweets (where \(N \ge 2\)),
of which \(a\) are red. Two sweets are drawn
from the bag without replacement. Show that
the probability that the first sweet is red
is equal to the probability that the second sweet is red.
There are two bags, each containing \(N\) sweets (where \(N \ge 2\)).
The first bag contains \(a\) red sweets, and the
second bag contains \(b\) red sweets. There is also a
biased coin, showing Heads with probability \(p\) and Tails with probability \(q\), where \(p+q = 1\).
The coin is tossed. If it shows Heads then a
sweet is chosen from the first bag and transferred
to the second bag; if it shows Tails then a sweet
is chosen from the second bag and transferred
to the first bag. The coin is then tossed a second time:
if it shows Heads then a sweet is chosen from the first bag,
and if it shows Tails then a sweet is chosen from the second bag.
Show that the probability that the first sweet
is red is equal to the probability that the second sweet is red.