Question Number 223055 by alcohol last updated on 13/Jul/25
Commented by alcohol last updated on 13/Jul/25
$${please}\:{i}\:{really}\:{really}\:{need}\:{a}\:{detailed} \\ $$$${explanation} \\ $$
Commented by mr W last updated on 15/Jul/25
$$\left.{part}\:\mathrm{2}\right)\:{of}\:{the}\:{question}\:{is}\:{not}\:{clear} \\ $$$${enough}.\:{is}\:{the}\:{correct}\:{code}\:{still}\:\mathrm{4321}? \\ $$
Answered by mr W last updated on 15/Jul/25
$$\left.\mathrm{1}\right) \\ $$$$\left.{a}\right)\: \\ $$$${number}\:{of}\:{four}\:{digit}\:{codes}\:{made}\:{up} \\ $$$${of}\:\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4}\:{is}\:\mathrm{4}!=\mathrm{24} \\ $$$$\left.{b}\right) \\ $$$${number}\:{of}\:{wrong}\:{codes}\:{is}\:\mathrm{24}−\mathrm{1}=\mathrm{23} \\ $$$${if}\:{a}\:{code}\:{has}\:{four}\:{wrong}\:{digits},\:{a} \\ $$$${alarm}\:{will}\:{be}\:{triggered}.\:{number}\:{of} \\ $$$${such}\:{codes}\:{is}\:!\mathrm{4}=\mathrm{9}. \\ $$$${for}\:{example}:\:\mathrm{4231}\:{is}\:{a}\:{wrong}\:{code}, \\ $$$${but}\:{it}\:{doesn}'{t}\:{trigger}\:{the}\:{alarm}. \\ $$$$\mathrm{3412}\:{is}\:{a}\:{wrong}\:{code}\:{which}\:{triggers} \\ $$$${the}\:{alarm}. \\ $$$$\left.{c}\right) \\ $$$${p}=\frac{\mathrm{9}}{\mathrm{24}}=\mathrm{37}.\mathrm{5\%} \\ $$$$ \\ $$$$\left.\mathrm{2}\right) \\ $$$$\left.{a}\right) \\ $$$${total}\:{number}\:{of}\:{possible}\:{tries}\:\mathrm{10}^{\mathrm{4}} \\ $$$${possibility}\:{that}\:{the}\:{door}\:{will}\:{open}\:{is} \\ $$$${p}=\frac{\mathrm{1}}{\mathrm{10}^{\mathrm{4}} } \\ $$$$\left.{b}\right) \\ $$$${possibility}\:{that}\:{the}\:{alarm}\:{will}\:{be} \\ $$$${triggered}\:{is} \\ $$$${p}=\frac{\mathrm{9}^{\mathrm{4}} }{\mathrm{10}^{\mathrm{4}} }=\mathrm{65}.\mathrm{61\%} \\ $$