SECTION 6.1 Discrete Random Variables 347
(a) Suppose that we wish to move along
a square grid (a 6 × 6 grid is
shown to the right) where we start
from the extreme northwest vertex
(A) and move toward the extreme
southeast vertex (B) in such a way
that we always move “toward” the
objective, i.e., each move is either
to the right (east) or down (south).
A moment’s thought reveals that there areÄ 12
6ä
such paths.
What is the probability that a random path from A to B
will always be above or on the diagonal drawn from A to
B? (Answer: For the grid to the right the probability is
C(6)/Ä 12
6ä
= 1/7.) This result generalizes in the obvious way
ton×ngrids.
(b) Suppose this time that we have 2n people, each wishing to
purchase a $10 theater ticket. Exactlyn of these people has
only a $10 bill, and the remainingn people has only a $20
bill. The person selling tickets at the ticket window has no
change. What is the probability that a random lineup of these
2 npeople will allow the ticket seller to make change from the
incoming receipts? (This means, for instance, that the first
person buying a ticket cannot be one of the people having
only a $20 bill.)- Suppose that we have a room with n politicians and that they
are going to use the following “democratic” method for selecting
a leader. They will distributen identical coins, each having the
probability of heads being p. The n politicians each toss their
respective coins in unison; if a politician’s coin comes up heads,
and if all of the others come up tails, then this politician becomes
the leader. Otherwise, they all toss their coins again, repeating
until a leader has been chosen.
(a) Show that the probability of a leader being chosen in a given
round isnp(1−p)n−^1.