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)\(P( ext{both red})\)
(b)\(P( ext{one of each})\)
(c)\(P( ext{second is blue})\)
(a)
Without replacement
\( frac{5}{8} imes frac{4}{7}= frac{20}{56}\)
(b)
RB or BR
\( frac{5}{8} imes frac{3}{7}+ frac{3}{8} imes frac{5}{7}= frac{15}{28}+ frac{15}{28}= frac{30}{56}\)
(c)
RB or BB
\( frac{5}{8} imes frac{3}{7}+ frac{3}{8} imes frac{2}{7}= frac{15}{56}+ frac{6}{56}\)
Question 2
A biased coin has \(P(H)= frac{2}{3}\). It is flipped three times. Find the probability of getting exactly two heads.
Three arrangements: HHT, HTH, THH
\(3 imes\left( frac{2}{3}
ight)^2 imes frac{1}{3}=3 imes frac{4}{27}\)
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 \(P( ext{red})\) and \(P( ext{Box A} \mid ext{red})\).
Total P(red)
\( frac{1}{2} imes frac{3}{5}+ frac{1}{2} imes frac{1}{5}= frac{3}{10}+ frac{1}{10}= frac{2}{5}\)
Bayes
\(P(A| ext{red})=\dfrac{3/10}{2/5}=\dfrac{3}{4}\)
Answer
\(P( ext{red})=\dfrac{2}{5}\); \(P(A| ext{red})=\dfrac{3}{4}\)
Generated by Mathski · mathski.io · IB Mathematics AA Shadow Worksheets