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Grade 10 Math · Problem type
Conditional probability on a Venn diagram
Use this for “Of 60 students, 25 drink juice, 30 milk and 10 both. What fraction of the juice drinkers drink milk?”
Diagram
Juice only 25 − 10 = 15
Milk only 30 − 10 = 20
Juice or Milk: 25 + 30 − 10 = 45 of 60
Shaded: P(Juice and Milk) = 10/60 ≈ 0.167
P(Milk | Juice) = 10/25 = 0.4: of the 25 in Juice, 10 are in Milk
This is an example. Type your own values to start a new problem.
P(Milk
|
Juice)
=
both
a
=
2
5
P(B | A)
b
Drink milk (B)
example
N
Students in all
example
P(A ∩ B)
P(A and B)
worked out (not typed)
P(A | B)
P(A | B)
worked out (not typed)
Formulas
P(A
∩
B)
=
both
N
1
6
=
10
60
P(B
|
A)
=
both
a
2
5
=
10
25
P(A
|
B)
=
both
b
1
3
=
10
30
Assumptions
•
Given A, only the A circle counts: P(B | A) is the overlap out of everyone in A, both ÷ a.
•
P(A | B) looks inside circle B instead: both ÷ b. The order of the events matters.
•
A and B are independent only when P(B | A) = P(B) = b ÷ the total.
Step-by-step
WE KNOW
a = 25, b = 30, both = 10, N = 60
FIND
P(A and B) (P(A ∩ B)), P(B | A) (P(B | A)), P(A | B) (P(A | B))
Step 1 · Find P(A and B): P(A ∩ B)
Use P(A ∩ B) = both ÷ N
The overlap out of everyone.
P(A
∩
B)
=
10
60
P(A
∩
B)
=
1
6
Step 2 · Find P(B | A): P(B | A)
Use P(B | A) = both ÷ a
Given A: only the people in A count, and “both” of them are in B.
P(B
|
A)
=
10
25
P(B
|
A)
=
2
5
Step 3 · Find P(A | B): P(A | B)
Use P(A | B) = both ÷ b
Given B: only the people in B count, and “both” of them are in A.
P(A
|
B)
=
10
30
P(A
|
B)
=
1
3
Check
Put the numbers back in. Both sides should match.
1
6
=
10
60
✓
2
5
=
10
25
✓
1
3
=
10
30
✓
Topic check
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