b To calculate the probabilities you have to consider the intersection of gender and clothing, as well as the intersection of gender, clothing, and shoes.
C
c The union of events requires either event happening. The intersection of events requires all events happening.
A
aTree Diagram:
B
bTree Diagram:
Probability of Long Pants: 12
C
cUnion:
male, long pants, tennis shoes
female, shorts, tennis shoes
female, long pants, tennis shoes
male, long pants, other shoes
female, long pants, other shoes
male, shorts, tennis shoes
female, dress or skirt, tennis shoes
male, long pants, other shoes
male, long pants, tennis shoes
Intersection:
male, long pants, tennis shoes
female, long pants, tennis shoes
male, long pants, tennis shoes
Practice makes perfect
a Let's start by identifying the unique parts of the descriptions in terms of sex, type of shoes, and clothing.
&male,long pants,tennis shoes
&female,shorts,tennis shoes
&female,dress or skirt,other shoes
&female,long pants,tennis shoes
&male,long pants,other shoes
&female,long pants,other shoes
&male,shorts,tennis shoes
&male,shorts,other shoes
&female,dress or skirt,tennis shoes
&male,long pants,other shoes
&female,shorts,other shoes
&male,long pants,tennis shoes
We have 2 genders, 3 different types of clothing, and 2 types of shoes. With this information, we can draw a tree diagram. Note that we cannot use an area model, as this can only show two events.
b To calculate probability we divide the number of favorable outcomes by the total number of outcomes.
P=Number of favorable outcomes/Number of possible outcomes
We know there are a total of 12 students. Let's summarize the number of cases of the different events.
ll
Male & 6
Female & 6
Long pants & 6
Shorts & 4
Dress/skirt & 2
Tennis shoes & 6
Other shoes & 6
The probability that a randomly selected student is wearing long pants is 612, which reduces to 12. To complete the second level of the diagram with probabilities, we have to consider the intersection of gender and the different types of clothing.
ll
Male, Long pants & 4
Male, Shorts & 2
Male, Mress/skirt & 0
Female, Long pants & 2
Female, Shorts & 2
Male, Mress/skirt & 2
Since there are 6 males and 6 females, we can complete the second level of the diagram with probabilities.
To complete the third level of the diagram, we have to consider the intersection of gender, the different types of clothing, and types of shoes.
ll
Male, Long pants, Tennis shoes & 2
Male, Long pants, Other shoes & 2
Male, Shorts, Tennis shoes & 1
Male, Shorts, Other shoes & 1
Male, Dress/skirt, Tennis shoes & 0
Male, Dress/skirt, Other shoes & 0
Female, Long pants, Tennis shoes & 1
Female, Long pants, Other shoes & 1
Female, Shorts, Tennis shoes & 1
Female, Shorts, Other shoes & 1
Female, Dress/skirt, Tennis shoes & 1
Female, Dress/skirt, Other shoes & 1
Since there are 6 males and 6 females, we can complete the second level of the diagram with probabilities.
c The union of long pants and tennis shoes includes all students who either wear long pants, or tennis shoes, or both. Let's list those.
&male,long pants,tennis shoes
&female,shorts,tennis shoes
&female,dress or skirt,other shoes
&female,long pants,tennis shoes
&male,long pants,other shoes
&female,long pants,other shoes
&male,shorts,tennis shoes
&male,shorts,other shoes
&female,dress or skirt,tennis shoes
&male,long pants,other shoes
&female,shorts,other shoes
&male,long pants,tennis shoes
As we can see, 9 outcomes are in the union of long pants and tennis shoes. The intersection of long pants and tennis shoes includes all students who wear both long pants and tennis shoes.
&male,long pants,tennis shoes
&female,shorts,tennis shoes
&female,dress or skirt,other shoes
&female,long pants,tennis shoes
&male,long pants,other shoes
&female,long pants,other shoes
&male,shorts,tennis shoes
&male,shorts,other shoes
&female,dress or skirt,tennis shoes
&male,long pants,other shoes
&female,shorts,other shoes
&male,long pants,tennis shoes
A total of 3 students are in the intersection of long pants and tennis shoes.
