Discrete Mathematics for Computer Science

Cross Product Sample Spaces 503

However, this event E 2 is part of a larger context in which a coin is flipped. In the grand

experiment^21 x Q2, getting an even number on the die corresponds to the event

{H, T} x {2, 4, 6} = {(H, 2), (H, 4), (H, 6), (T, 2), (T, 4), (T, 6)}

because either H or T could have turned up on the coin flip. In other words, the nonempty

event Ei in Q^2 i corresponds to the event

E* = Q21 X Q2 X ... X Q'i-1 x Ei x K'2i4_1 X ... X Q'n

in ý2 = Q1 X 02 X ... x Q,. The following corollary of Theorem 3 shows that this re-

formulation to a corresponding event preserves the probability computed for the original

event.

Corollary 1 to Theorem 3. P (E7) = P (Ei) where

E*ý = Q1 x ... x Qi-1 x Ei x Qi+I x ... x 0,2

is an event in cross product sample space Q1 X 02 X ... X Qn.

Proof. According to Theorem 3,

P(E7) = P(Q2 1 ) • P(Q 2 ) ... P(A2i- 1 ) • P(Ei) .P(Qi+I) ... P(Q2n)

Each "j has P(Qj) = 1, so P(E7) = P(Ei). U

Example 6. An experiment involves flipping a fair coin and rolling a fair die. Let Q =

{H, T} x {1, 2, 3, 4, 5,^61 be a cross product sample space for the experiment. Compute

the probability of rolling a 3 or a^5 as an event in^0 using Corollary^1 to Theorem 3.

Solution. Define sample spaces Q1I = {H, T} and Q22 = {1, 2, 3, 4, 5, 6} so that Q =
01 x Q2. The event of interest in Q2 is E= {(H, 3), (T, 3), (H, 5), (T, 5)1. The event of
interest in "22 is E 2 = {3, 51, which has probability 2/6 = 1/3. Since iJQI = 12, P(E) =
4/12 = 1/3. According to Corollary 1 to Theorem 3, compute P(E) = P(E 2 ), which is,
indeed, the case in this example. M
We can also specify events in a cross product sample space by specifying what takes
place in some collection of individual sample spaces. Suppose we single out k sample
spaces^2 il, I"Oi2, • -I "ik from the n sample spaces Q"1, •2•2 ... 2n where k < n, and then
specify an event Eij C £jj, in each of these k selected sample spaces. The events Eij cor-
respond to events EV Iiin the cross product Q21 x ... X On- Specifying that all the events

El ...... Eik are to occur in the individual sample spaces is the same thing as specify-

ing that the event E. n •... n E* is to occur in the cross product sample space. The next
corollary of Theorem 3 says that the probability of this event can be computed by a multi-
plication rule.
Corollary 2 to Theorem 3. P(E• nEýn ... NEa) = P(Ei,). P(Ei 2 ) ... P(Eik),

where

Ef = ia x ... x Eij x ... x Q,

for I < j < k is an event in cross product sample space Q,1 X Q2 x .. • X Qn.