Cross Product Sample Spaces
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However, this event E 2 is part of a larger context in which a coin is flipped. In the grand
experiment
2 1 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 reformulation 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).
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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.
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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 correspond to events EV in the cross product Q21 x ... X On- Specifying that all the events
Ii
El ...... Eik are to occur in the individual sample spaces is the same thing as specifying 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 multiplication 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.