Question Number 12506 by Joel577 last updated on 24/Apr/17
$$\mathrm{A}\:\mathrm{wooden}\:\mathrm{stick}\:\mathrm{was}\:\mathrm{broken}\:\mathrm{randomly}\:\mathrm{into} \\ $$$$\mathrm{three}\:\mathrm{pieces}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{a}\:\mathrm{triangle} \\ $$$$\mathrm{can}\:\mathrm{be}\:\mathrm{built}\:\mathrm{from}\:\mathrm{those}\:\mathrm{three}\:\mathrm{parts}? \\ $$
Answered by mrW1 last updated on 25/Apr/17
$${Let}'{s}\:{say}\:{the}\:{wooden}\:{stick}\:{is}\:{brocken}\: \\ $$$${into}\:{three}\:{pieces}\:{A},\:{B},{C}. \\ $$$${The}\:{length}\:{of}\:{wooden}\:{stick}\:{is}\:\mathrm{1}. \\ $$$$ \\ $$$${When}\:{a}\:{wooden}\:{stick}\:{is}\:{brocken}\:{into} \\ $$$$\mathrm{3}\:{pieces},\:{there}\:{are}\:\mathrm{4}\:{possibilities}: \\ $$$$\left.\mathrm{1}\right)\:{none}\:{of}\:{the}\:{three}\:{pieces}\:{is}\:{longer} \\ $$$${than}\:{or}\:{equal}\:{to}\:\mathrm{1}/\mathrm{2}; \\ $$$$\left.\mathrm{2}.\mathrm{1}\right)\:{one}\:{of}\:{the}\:{three}\:{pieces}\:{is}\:{longer} \\ $$$${than}\:{or}\:{equal}\:{to}\:\mathrm{1}/\mathrm{2},\:{and}\:{this}\:{piece}\:{is}\:{A}; \\ $$$$\left.\mathrm{2}.\mathrm{2}\right)\:{one}\:{of}\:{the}\:{three}\:{pieces}\:{is}\:{longer} \\ $$$${than}\:{or}\:{equal}\:{to}\:\mathrm{1}/\mathrm{2},\:{and}\:{this}\:{piece}\:{is}\:{B}; \\ $$$$\left.\mathrm{2}.\mathrm{3}\right)\:{one}\:{of}\:{the}\:{three}\:{pieces}\:{is}\:{longer} \\ $$$${than}\:{or}\:{equal}\:{to}\:\mathrm{1}/\mathrm{2},\:{and}\:{this}\:{piece}\:{is}\:{C}. \\ $$$$ \\ $$$$\left.{Only}\:{in}\:{case}\:\mathrm{1}\right)\:{the}\:{three}\:{pieces}\:{can} \\ $$$${build}\:{a}\:{triangle}. \\ $$$$\Rightarrow{probability}\:=\mathrm{1}/\mathrm{4}. \\ $$
Answered by ajfour last updated on 24/Apr/17
$$\bullet\:\:{break}\:{the}\:{stick}\:{anywhere} \\ $$$$\bullet\:{for}\:{success}\:{break}\:{the}\:{larger}\: \\ $$$$\:\:\:\:{of}\:{the}\:{two}\:{pieces} \\ $$$${so}\:{probability}\:{that}\:{a}\:{triangle} \\ $$$${is}\:{formed}\:=\:\mathrm{1}×\left(\mathrm{1}/\mathrm{2}\right)\:. \\ $$
Commented by mrW1 last updated on 25/Apr/17
$${what}\:{if}\:{the}\:{stick}\:{is}\:{broken}\:{into}\:{two} \\ $$$${equal}\:{pieces}? \\ $$$$ \\ $$$${break}\:{the}\:{larger}\:{of}\:{the}\:{two}\:{pieces}\:{doesnt} \\ $$$${always}\:{mean}\:{success},\:{e}.{g}. \\ $$$${break}\:\mathrm{1}\:{into}:\:\mathrm{1}/\mathrm{5},\:\mathrm{4}/\mathrm{5} \\ $$$${break}\:\mathrm{4}/\mathrm{5}\:{into}:\:\mathrm{1}/\mathrm{5},\:\mathrm{3}/\mathrm{5} \\ $$$$\mathrm{1}/\mathrm{5},\:\mathrm{1}/\mathrm{5}\:{and}\:\mathrm{3}/\mathrm{5}\:{can}\:{not}\:{build}\:{a}\:{triangle}. \\ $$
Commented by ajfour last updated on 25/Apr/17
$${thanks}\:{for}\:{the}\:{light}. \\ $$