Answer to Question #127601 in Statistics and Probability for Harsha

Question

In a certain group of 15 students, 5 have graphics calculators and 3 have a Laptop at home (one student has
both). Two of the students drive themselves to campus each day and neither of them has a graphics calculator nor
a Laptop at home. A student is selected at random from the group.
((a) Find the probability that the student either drives to campus or has a graphics calculator.
(b) Show that the events “the student has a graphics calculator” and “the student has a Laptop at home” are
independent.
Let G represent the event “the student has a graphics calculator”
H represent the event “the student has a Laptop at home”
D represent the event “the student drives to campus each day”
Represent the information in this question by a Venn diagram. Use the above Venn diagram to answer the questions.

Answer 1

$|U|=15, |G|=5, |H|=3, |G \cap H|=1, |D|=2, G \cap D = \varnothing, H \cap D = \varnothing \\[0.2cm]$

(a)

Probability is an amount of drivers or students with calculator divided by amount of all students

$P(D \cup G)=\dfrac{|D \cup G|}{|U|} = \dfrac{|D|+|G|-|D \cap G|}{|U|}=\dfrac{2+5}{15} = \dfrac{7}{15} \approx 0.467$

(b)

Events are independent if $P(G \cap H)=P(G)\cdot P(H)\\[0.1cm]$

$P(G) = \dfrac{5}{15}$, $P(H) = \dfrac{3}{15}$. So $P(G)\cdot P(H) = \dfrac{15}{225} = \dfrac{1}{15}. \\[0.1cm]$

$P(G \cap H)=\dfrac{1}{15}.$

Events are independent. QED.