Solution 2 the easy way suppose that alice first

2

Problem 3 (p.31 #5 a, d) . Let Ω be the sample space corresponding to three tosses of a coin. Give
a verbal description of the following events:
a) A = { HHH, HHT, HTH, HTT } ;
b) B = { HHH, HHT, HTH, THH } ;
c) Show that A and B are not independent.
d) Find a third event, C , so that P ( C ) =
1
2
and P ( A ∩ B ∩ C ) = P ( A ) P ( B ) P ( C ). So the events
A, B, C are not independent, but they have this property.
3

Problem 4 (p.31 #11) . Let A, B, and C be three events. Use the inclusion-exclusion formula for
two events to derive the following inclusion-exclusion formula for three events:
P ( A ∪ B ∪ C ) = P ( A ) + P ( B ) + P ( C ) - P ( AB ) - P ( AC ) - P ( BC ) + P ( ABC ) .