SME
Grade 10 Math · Geometry
Conditional probability and independence
Use this for “From the two-way table, what is the chance a student who rides the bus is late?”
Table
160 in all: Late 25, On time 135
Conditional relative frequency: P(Late | Bus) = 12/60 = 0.2, out of the Bus total only
This is an example. Type your own values to start a new problem.
P(Late
|
Bus)
=
LB
B
=
1
5
P(Late | Bus)
LW
Late, walk
example
LC
Late, car
example
OB
On time, bus
example
OW
On time, walk
example
OC
On time, car
example
L
Late total
worked out (not typed)
P(Bus | Late)
P(Bus | Late)
worked out (not typed)
Formulas
B
=
LB
+
OB
60 = 12 + 48
P(Late
|
Bus)
=
LB
B
1
5
=
12
60
L
=
LB
+
LW
+
LC
25 = 12 + 4 + 9
P(Bus
|
Late)
=
LB
L
12
25
=
12
25
Assumptions
•
P(Late | Bus) looks only at the bus column: those late out of everyone who took the bus.
•
The whole is the column total, not the grand total.
•
P(Bus | Late) looks at the late row instead: the order of the events matters.
Step-by-step
WE KNOW
LB = 12, LW = 4, LC = 9, OB = 48, OW = 36, OC = 51
FIND
bus total (B), P(Late | Bus) (P(Late | Bus)), late total (L), P(Bus | Late) (P(Bus | Late))
Step 1 · Find bus total (B)
Use B = LB + OB
Add the bus column.
B
=
12
+
48
B
=
60
Step 2 · Find P(Late | Bus): P(Late | Bus)
Use P(Late | Bus) = LB ÷ B
Late among the bus riders only: the cell over its column total.
P(Late
|
Bus)
=
12
60
P(Late
|
Bus)
=
1
5
Step 3 · Find late total (L)
Use L = LB + LW + LC
Add the late row.
L
=
12
+
4
+
9
L
=
25
Step 4 · Find P(Bus | Late): P(Bus | Late)
Use P(Bus | Late) = LB ÷ L
Bus riders among the late students: the cell over its row total.
P(Bus
|
Late)
=
12
25
P(Bus
|
Late)
=
12
25
Check
Put the numbers back in. Both sides should match.
60 = 12 + 48 ✓
1
5
=
12
60
✓
25 = 12 + 4 + 9 ✓
12
25
=
12
25
✓
Topic check
See what you know: 5 questions on this topic
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