MAST30001 Stochastic Modelling
Tutorial Sheet 1
1. A box has 3 drawers, one contains two gold coins, one contains two silver coins and
the last drawer contains one gold coin and one silver coin. Assume that one drawer
is selected randomly and that a randomly selected coin from that drawer turns out
to be gold. What is the probability that the chosen drawer is the one that contains
the two gold coins?
Let Ai be the event that the drawer with i gold coins is chosen, i = 0, 1, 2, and G be
the event that a gold coin is chosen.
P(A2|G) =
P(A2 ∩G)
P(G)
=
P(A2)
P(G)
=
1/3
1/2
=
2
3
2. Let (Ai)i∈Z+ be events, and suppose that P(∩
n−1
i=0 Ai) > 0. Show that
P(∩ni=0Ai) = P(A0)
n∏
m=1
P(Am| ∩m−1i=0 Ai).
This can be proved by induction on n, using P(∩ni=0Ai) = P(An| ∩
n−1
i=0 Ai)P(∩
n−1
i=0 Ai)
etc., but one can also see it directly: By the definition of conditional probability (all
of them are well defined), for each m ≤ n
P(Am| ∩m−1i=0 Ai) =
P(∩mi=0Ai)
P(∩m−1i=0 Ai)
.
Thus when we write out the product of them, lots of things cancel and we get
n∏
m=1
P(Am| ∩m−1i=0 Ai) =
P(∩ni=0Ai)
P(A0)
.
3. A game involves rolling a ball around a table, starting from a fixed position. Three
players, Alex, Bobby and Célia, take turns attempting to grab the ball as it passes
them. If Alex is unsuccessful then Bobby attempts to grab the ball and if he is also
unsuccessful then Célia attempts to grab the ball. If they are all unsuccessful, the
ball is rolled again from the starting position. The game stops as soon as any player
is successful (and that player wins!). Suppose that each player is able to grab the
passing ball with probability p ∈ (0, 1), independent of previous attempts.
(a) What is the distribution of the number of grabbing attempts until the game
ends?
Geometric(p).
(b) Find the probability that all 3 players fail to grab the ball on their first at-
tempt.
(1− p)3.
(c) In one game, what is the distribution of the number of times that the ball is
rolled from the starting position?
Geometric(1− (1− p)3).
(d) Find the probability that Alex wins the game.
p+ (1− p)3p+ (1− p)6p+ · · · = p
∞∑
n=0
(
(1− p)3
)n
=
p
1− (1− p)3
.
(e) Find the probability that Bobby wins the game.
(1− p)p+ (1− p)4p+ (1− p)7p+ · · · = p(1− p)
∞∑
n=0
(
(1− p)3
)n
=
p(1− p)
1− (1− p)3
.
(f) Find the probability that Célia wins the game.
(1−p)2p+(1−p)5p+(1−p)8p+ · · · = p(1−p)2
∞∑
n=0
(
(1−p)3
)n
=
p(1− p)2
1− (1− p)3
.
Note that you can also get this from the fact that
p+ p(1− p) + p(1− p)2 = 3p− 3p2 + p3 = 1− (1− p)3.
(g) If this game is played 3 times, what is the distribution of the number of games
that Alex wins?
The number of games X that Alex wins has a Binomial(3, p
1−(1−p)3 ) distribution
4. Let X1, X2, . . . be independent random variables with Xi ∼geometric(pi) (each tak-
ing values in N).
(a) Show that P(X1 > n+m|X1 > m) = P(X1 > n) (for n,m ∈ N).
Since
P(X1 > n+m|X1 > m) =
P(X1 > n+m,X1 > m)
P(X1 > m)
=
P(X1 > n+m)
P(X1 > m)
,
We see that the LHS of the claimed equality is equal to (1−p1)n+m/(1−p1)m =
(1− p1)n, which is the same as the right hand side.
(b) Find the distribution of Yn = mini≤nXi.
Yn also takes values in N. Moreover, for x ∈ N,
P(Yn > x) = P(∩ni=1{Xi > x}) =
n∏
i=1
P(Xi > x) =
n∏
i=1
(1− pi)x
=
(
n∏
i=1
(1− pi)
)x
.
Therefore, Yn ∼geometric(γn), where γn = (1−
∏n
i=1(1− pi)).
(c) Find the probability mass function of the random vector (Yn, Yn+1).
Clearly this vector has components in N. Moreover Yn+1 ≤ Yn by definition,
thus P(Yn = m,Yn+1 = k) = 0 if k > m. If k = m then
P(Yn = m,Yn+1 = m) = P(Yn = m,Xn+1 ≥ m) = P(Yn = m)P(Xn+1 ≥ m)
= (1− γn)m−1γn(1− pn+1)m−1
If k < m then P(Yn = m,Yn+1 = k) = P(Yn = m,Xn+1 = k) = P(Yn = m)P(Xn+1 = k) = (1− γn)m−1γn(1− pn+1)k−1pn+1. (d) Find E[Yn+1|Yn = m]. Note that P(Yn+1 = k|Yn = m) = P(Xn+1 = k), if k = 1, . . . ,m− 1 P(Xn+1 ≥ m), if k = m 0, otherwise. It follows that E[Yn+1|Yn = m] = mP(Xn+1 ≥ m) + m−1∑ k=1 kP(Xn+1 = k) = m(1− pn+1)m−1 + m−1∑ k=1 k(1− pn+1)k−1pn+1.