Introduction to Statistics · Example 3.3.5 About one woman in seven who lives to be 90 will develop breast cancer. Of those women who develop breast cancer, a test is negative 2% of the time. In the general population of women, the test is negative about 85% of the time.
B = woman develops breast cancer  ·  N = tests negative  ·  one woman chosen at random
1 in 7
2% and 85%
Two rates. Two different groups of women.
B = develops cancer   N = tests negative
Part a — the unconditional rates
one in seven = 1 ÷ 7 = 0.143
P(B) = 0.143
85% = 85 ÷ 100 = 0.85
P(N) = 0.85
Part b — the 2% is not what it looks like
“Of those women who develop breast cancer…”
all 1,000 → only the 143 with B
2% = 2 ÷ 100 = 0.02
P(N | B) = 0.02
P(N) = 0.85  vs  P(N | B) = 0.02
The bar is not decoration. It changes who you are talking about.
Part c — both things happen
P(B AND N) = P(B) · P(N | B)
all women → 14.3% → 2% of those
(0.143)(0.02) = 0.0029
≈ 3 women in 1,000 — count them.
Part d — one thing or the other
P(B OR N) = P(B) + P(N) − P(B AND N)
0.143 + 0.85 − 0.0029 = 0.9901
Part e — are B and N independent?
0.85 vs 0.02
P(N | B) ≠ P(N) → not independent
A test whose result didn’t depend on the disease would be useless. The dependence is the diagnostic power.
Part f — can both happen at once?
P(B AND N) = 0.0029 ≠ 0 → not mutually exclusive
That is a false negative — and we just measured how often it happens.
All six answers
0.143 · 0.85 · 0.02 · 0.0029 · 0.9901 · no · no
Six words changed which group is under discussion. When one of these seems intractable, look for that clause first.
  • Each dot is one woman — 1,000 of them.
  • 143 of the 1,000 develop breast cancer.
  • Of those 143, the test misses 2%.
  • That is 3 women in 1,000 — the three blue dots.
  • Those three blue dots are the false negatives.

Answers so far

  • part a — P(B)0.143
  • part a — P(N)0.85
  • part b — P(N | B)0.02
  • part c — P(B AND N)0.0029
  • part d — P(B OR N)0.9901
  • part e — independent?no
  • part f — mutually exclusive?no