Probability III: Conditional Probability
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
In a year group, 55% enjoy Sport (S) and 70% enjoy Music (M). 30% enjoy neither. A student is chosen at random.
(a)Use a Venn diagram to find \(P(S\cap M)\).
(b)Given a student enjoys Music, find the probability they also enjoy Sport.
(c)Given a student enjoys at least one activity, find the probability they enjoy both.
(a)
Total in at least one
\(1-0.30=0.70\); \(P(S\cap M)=0.55+0.70-0.70=0.55\)
Answer
\(P(S\cap M)=0.55\) — wait: \(P(S)+P(M)-P(S\cap M)=0.70\Rightarrow P(S\cap M)=0.55\)
(b)
Conditional
\(P(S|M)=\dfrac{P(S\cap M)}{P(M)}=\dfrac{0.55}{0.70}\)
Answer
\(\dfrac{11}{14}\approx0.786\)
(c)
Answer
\(\dfrac{0.55}{0.70}=\dfrac{11}{14}\)
Question 2
Two dice are rolled. Let A be "sum is 8" and B be "first die shows 3". Find \(P(A|B)\) and \(P(B|A)\).
\(P(A\cap B)\)
First die 3, second die 5: 1 outcome. \(P(A\cap B)=\tfrac{1}{36}\)
\(P(A)=\tfrac{5}{36}\) (sums: 2+6,3+5,4+4,5+3,6+2); \(P(B)=\tfrac{6}{36}=\tfrac{1}{6}\)
Answer
\(P(A|B)=\dfrac{1/36}{1/6}=\dfrac{1}{6}\); \(P(B|A)=\dfrac{1/36}{5/36}=\dfrac{1}{5}\)
Question 3
The Venn diagram shows \(P(A\text{ only})=0.3\), \(P(A\cap B)=0.2\), \(P(B\text{ only})=0.1\). Find \(P(A|B')\).
\(P(B')=1-P(B)=1-0.3=0.7\)
\(P(A\cap B')=0.3\) (A only region)
Answer
\(P(A|B')=\dfrac{0.3}{0.7}=\dfrac{3}{7}\)
Question 4
A factory has two machines, X and Y. X produces 60% of items, Y produces 40%. X has a 3% defect rate, Y has a 5% defect rate. Find the probability that a randomly chosen defective item came from machine X.
Bayes' theorem
\(P(D)=0.6\times0.03+0.4\times0.05=0.018+0.02=0.038\)
\(P(X|D)\)
\(\dfrac{0.018}{0.038}=\dfrac{9}{19}\)
Answer
\(\dfrac{9}{19}\approx0.474\)
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