Tree Diagrams
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
A bag contains 5 red and 3 blue balls. Two balls are drawn without replacement. Draw a tree diagram and find:
(a)\(P(\text{both red})\)
(b)\(P(\text{one of each})\)
(c)\(P(\text{second is blue})\)
(a)
Without replacement
\(\tfrac{5}{8}\times\tfrac{4}{7}=\tfrac{20}{56}\)
(b)
RB or BR
\(\tfrac{5}{8}\times\tfrac{3}{7}+\tfrac{3}{8}\times\tfrac{5}{7}=\tfrac{15}{28}+\tfrac{15}{28}=\tfrac{30}{56}\)
(c)
RB or BB
\(\tfrac{5}{8}\times\tfrac{3}{7}+\tfrac{3}{8}\times\tfrac{2}{7}=\tfrac{15}{56}+\tfrac{6}{56}\)
Question 2
A biased coin has \(P(H)=\tfrac{2}{3}\). It is flipped three times. Find the probability of getting exactly two heads.
Three arrangements: HHT, HTH, THH
\(3\times\left(\tfrac{2}{3}\right)^2\times\tfrac{1}{3}=3\times\tfrac{4}{27}\)
Question 3
Box A has 3 red and 2 blue balls. Box B has 1 red and 4 blue balls. A box is chosen at random, then a ball drawn. Find \(P(\text{red})\) and \(P(\text{Box A} \mid \text{red})\).
Total P(red)
\(\tfrac{1}{2}\times\tfrac{3}{5}+\tfrac{1}{2}\times\tfrac{1}{5}=\tfrac{3}{10}+\tfrac{1}{10}=\tfrac{2}{5}\)
Bayes
\(P(A|\text{red})=\dfrac{3/10}{2/5}=\dfrac{3}{4}\)
Answer
\(P(\text{red})=\dfrac{2}{5}\); \(P(A|\text{red})=\dfrac{3}{4}\)
Generated by Mathski · mathski.io · IB Mathematics AA Shadow Worksheets