Question

A survey of 300 people found that 60 owned an iPhone, 75 owned a Blackberry, and 30 owned an Android phone. Furthermore, 40 owned both an iPhone and a Blackberry, 12 owned both an iPhone and an Android phone, and 8 owned a Blackberry and an Android phone. Finally, 3 owned all three phones. (a) How many people surveyed owned none of the three phones? (b) How many people owned a Blackberry but not an iPhone? (c) How many owned a Blackberry but not an Android?

Answer 1

Step-by-step explanation:

We let A be the set of those who owned an iPhone, B be the set of those who owned a Blackberry,

and C those that owned an Android. Therefore we have |A| = 60, |B| = 75, and |C| = 30. Also,

we know that |A ∩ B| = 40, |A ∩ C| = 12 and |B ∩ C| = 8. Finally, |A ∩ B ∩ C| = 3. Therefore,

by the principle of inclusion/exclusion, we have

|A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|

= 60 + 75 + 30 − 40 − 12 − 8 + 4

= 109

(a) Since there are 300 total people, there are 191 without one of the types of phone.

(b) Since |A ∩ B| = 40 and |B| = 75, there are 35 people with a Blackberry that don’t own

an iPhone.

(c) Since |B ∩ C| = 8 and |B| = 75, there are 67 people with a Blackberry that don’t own

an Android.

It is also helpful to draw Venn diagrams for this problem.