Construction of Venn
Diagrams:
With three overlapping events
Country western songs emphasize three basic themes:
love, prison, and trucks. A survey of the local country
western radio station produced the following data:
45% of songs are about prisoners
50% of songs are about truck drivers
57% of songs are about love
28% of songs are about truck drivers in prison
20% of songs are about truck drivers in love
25% of songs are about prisoners in love
11% of songs are about all three
We will answer some probability questions and will
need a Venn diagram to make the problem easier.
We can now start to fill in probabilities as we find them.
It is easiest if we start with the innermost region. This
corresponds to songs about prison, trucks, and love.
Now we look at combinations of two events. 28% of songs
are about truck drivers in prison, and we have 11%
already accounted for in the innermost region. This
leaves 17% for the remaining region representing songs
of truckers in prison,but not in love.
The next combination is 20% with truckers and love.
With 11% already accounted, we have 9% in the region
for love, trucks, but not prison.
The next combination is 25% about prisoners in love.
With 11% already accounted, we have 14% in the
region for love and prison, but not trucks.
Now with 45% of the songs about prisoners, we see
that 3% are about prison only.
50% of the songs about truckers, we see that 13% are
about trucks only, and with 57% about love, we still
need another 23% that will be love only.
This leaves us only with the possibility of songs that are
not about love, prison, or trucks. Hard to believe, right?
Adding up everything we have, so far, gives 90%, so
10% of the country western songs are about something
else!
Now we can answer some questions.
For a song selected at random what is the probability
that the song is about a truck driver who is not in prison
but is in love?
P=.09
For a song selected at random what is the probability
that the song is not about trucks, prison, or love?
P=.10
For a song selected at random what is the probability
that the song is about truck drivers or prisoners?
P=.67
What is the probability of a song being about truck
drivers, given that it is about love?
P=(.09+.11)/.57=.20/.57=.35