Question

olitical analysts estimate the probability that Hillary Clinton will run for president in
2008 is 45%, and the probability that NY's Governor George Pataki will run as the Republican
candidate is 20%. If their political decisions are independent, then what is the probability that
only Hillary runs for president?

Answer 1

Let H = Event that Hillary runs
Let P = Even that Pataki runs
The probability that Hillary runs is
$$
H=H\cap(P\cup\bar{P})\\
\text{Therefore, }\\
=P((H\cap P)\cup (H\cap\bar P))\\
=P(H|P)P(P)+P(H|\bar P)P(\bar P)\\
=P(H)P(P)+P(H)P(\bar P)\\
=(0.45)(0.20)+(0.45)(0.80)\\
=0.45
$$
Of course.
Since the probability that Hillary runs is independent of whether or not Pataki runs.

Answer 2

the answer is .36

Answer 3

Okay.

The probability that ONLY Hilary runs is the same as the probability that Hilary runs and Pitaki does not run
$$
P(H\cap \bar P)=P(H)P(\bar P)=0.45\times0.80=0.36
$$