Question

3-24 If E and F are two mutually exclusive events, what can be said about their independence? [Hint: What is Pr(E and F)? Then, using (3-17), what is Pr(E|F)? Does it equal Pr(E)?]

Answer 1

24) If and are mutually exclusive events, then
PENF= 0
.

So,

P (EnF ) 0 P (EF) P(F) 0

That is
PEFΟ
unless $P\left ( F \right )=0$ . Thus,

$P\left ( E|F \right )=0\neq P(E)$

So and are not independent.

17) Assume there is an equal chance of boys and girls. $P\left ( B \right )=P(G)=0.5$

a) The probability of no girls (all 3 boys) is

$P\left ( BBB \right )=0.5^3=0.125$

So, use complement

$P\left ( \textup{at least one girl}\right )=1-0.125={\color{Blue} 0.875}$

b) The probability of at least 2 girls is

$P\left ( \textup{at least 2 girls}\right )=P(GGB)+P(GBG)+P(BGG)+P(GGG)\\ P\left ( \textup{at least 2 girls}\right )=0.5^3+0.5^3+0.5^3+0.5^3\\ {\color{Blue} P\left ( \textup{at least 2 girls}\right )=0.5}$

c) The required conditional probability is

$P\left ( \textup{at least 2 girls}|\textup{at least 1 girl}\right )=\frac{P\left ( \textup{at least 2 girls}\cap \textup{at least 1 girl}\right )}{P\left ( \textup{at least 1 girl}\right )}\\ P\left ( \textup{at least 2 girls}|\textup{at least 1 girl}\right )=\frac{P\left ( \textup{at least 2 girls}\right )}{P\left ( \textup{at least 1 girl}\right )}\\ P\left ( \textup{at least 2 girls}|\textup{at least 1 girl}\right )=\frac{0.5}{0.875}\\ {\color{Blue} P\left ( \textup{at least 2 girls}|\textup{at least 1 girl}\right )=0.5714}$

d) The required conditional probability is

$P\left ( \textup{at least 2 girls}|\textup{ eldest child is a girl }\right )=P(GBG)+P(BGG)+P(GGG)\\ P\left ( \textup{at least 2 girls}|\textup{ eldest child is a girl }\right )=0.5^3+0.5^3+0.5^3\\{\color{Blue} P\left ( \textup{at least 2 girls}|\textup{ eldest child is a girl }\right )=0.375}$