SME
Grade 10 Math · Problem type
Neither event
Use this for “Of 40 students, 22 play soccer, 15 basketball and 8 both. How many play neither, and what is the chance of neither?”
Diagram
Soccer only 22 − 8 = 14
Basketball only 15 − 8 = 7
Soccer or Basketball: 22 + 15 − 8 = 29 of 40
Neither: 40 − 29 = 11, outside both circles
Shaded: P(neither) = 11/40 = 0.275
This is an example. Type your own values to start a new problem.
Neither:
s
=
N
−
(
a
+
b
−
both
)
u
In Soccer or Basketball
worked out (not typed)
P(neither)
P(neither)
worked out (not typed)
Formulas
u
=
a
+
b
−
both
29 = 22 + 15 − 8
s
=
N
−
u
11 = 40 − 29
P(neither)
=
s
N
11
40
=
11
40
Assumptions
•
Each circle counts the people in it; the overlap counts those in both.
•
Soccer or basketball counts the overlap once: a + b − both.
•
Neither is everyone else in the rectangle: the total minus that.
Step-by-step
WE KNOW
a = 22, b = 15, both = 8, N = 40
FIND
in Soccer or Basketball (u), in neither (s), P(neither) (P(neither))
Step 1 · Find in Soccer or Basketball (u)
Use u = a + b − both
Add the two circles, then take off the overlap counted twice.
u
=
22
+
15
−
8
u
=
29
Step 2 · Find in neither (s)
Use s = N − u
Everyone not in either circle.
s
=
40
−
29
s
=
11
Step 3 · Find P(neither): P(neither)
Use P(neither) = s ÷ N
The count in neither over everyone.
P(neither)
=
11
40
P(neither)
=
11
40
Check
Put the numbers back in. Both sides should match.
29 = 22 + 15 − 8 ✓
11 = 40 − 29 ✓
11
40
=
11
40
✓
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