Question

Let S be a sample space and E and F be events associated with S. Suppose that Pr (Upper E )equals 0.6​, Pr (Upper F )equals 0.2 and Pr (Upper E intersect Upper F )equals 0.1. Calculate the following probabilities. ​(a) Pr (E|F )​(b) Pr (F|E )​(c) Pr (E| Upper F prime )​(d) Pr (Upper E prime | Upper F prime )

Answer 1

Answer:

(a)0.5

(b)0.17

(c)0.625

(b)0.375

Step-by-step explanation:

Pr(E)=0.6​

Pr(F)=0.2

$Pr(E\cap F)=0.1.$

(a)Pr (E|F )

$Pr (E|F )=\dfrac{Pr(E \cap F)}{Pr(F)} \\=\dfrac{0.1}{0.2}\\\\=0.5$

(b)Pr (F|E )

$Pr (F|E )=\dfrac{Pr(E \cap F)}{Pr(E)} \\=\dfrac{0.1}{0.6}\\\\=0.17$

(c)Pr (E|F')​

Pr(F')=1-P(F)

=1-0.2=0.8

$Pr(E \cap F')=P(E)-P(E\cap F)\\=0.6-0.1\\=0.5$

Therefore:

$Pr (E|F' )=\dfrac{Pr(E \cap F')}{Pr(F')} \\=\dfrac{0.5}{0.8}\\\\=0.625$

(d)Pr(E'|F')​

$P(E'\cap F')=P(E \cup F)'\\=1-P(E \cup F)\\=1-[P(E)+P(F)-P(E\cap F)]\\=1-[0.6+0.2-0.1]\\=1-0.7\\=0.3$

Therefore:

$Pr (E'|F' )=\dfrac{Pr(E' \cap F')}{Pr(F')} \\=\dfrac{0.3}{0.8}\\\\=0.375$