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Category: Probability and Statistics

three-points-are-randomly-selected-on-a-circle-to-form-a-triangle-1-find-the-probability-that-the-center-of-the-circle-lies-inside-the-triangle-2-find-the-probability-that-the-triangle-is-an-acu

Question Number 209510 by mr W last updated on 12/Jul/24 $${three}\:{points}\:{are}\:{randomly}\:{selected} \\ $$$${on}\:{a}\:{circle}\:{to}\:{form}\:{a}\:{triangle}.\: \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{probability}\:{that}\:{the}\:{center} \\ $$$${of}\:{the}\:{circle}\:{lies}\:{inside}\:{the}\:{triangle}. \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{probability}\:{that}\:{the}\: \\ $$$${triangle}\:{is}\:{an}\:{acute}\:{triangle}. \\ $$ Terms of…

Question-209424

Question Number 209424 by Tawa11 last updated on 09/Jul/24 Commented by Tawa11 last updated on 09/Jul/24 $$\mathrm{In}\:\mathrm{this}\:\mathrm{kind}\:\mathrm{of}\:\mathrm{probability},\:\mathrm{is}\:\mathrm{the}\:\mathrm{formular}. \\ $$$$\mathrm{p}\left(\mathrm{x}\:=\:\mathrm{r}\right)\:\:=\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{r}} \:\mathrm{p}^{\mathrm{r}} \:\mathrm{q}^{\mathrm{n}\:−\:\mathrm{r}} \\ $$$$\mathrm{or} \\…

Question-209031

Question Number 209031 by Spillover last updated on 30/Jun/24 Answered by Spillover last updated on 07/Jul/24 $${f}\left({x}\right)=\begin{cases}{\frac{\mathrm{1}}{{n}}\:\:\:\:{x}=\mathrm{1},\mathrm{2},\mathrm{3},…..=\frac{\mathrm{1}}{{n}}\left[{e}^{{t}} +{e}^{\mathrm{2}{t}} +{e}^{\mathrm{3}{t}} +……..\right]}\\{\mathrm{0}\:\:{else}\:{where}}\end{cases} \\ $$$$\left({a}\right)\:{moment}\:{generating}\:{function} \\ $$$${M}_{{x}} \left({t}\right)={E}\left({e}^{{tx}}…

n-married-couples-are-invited-to-a-dance-party-for-the-first-dance-n-paires-are-radomly-selected-what-s-the-probability-that-no-woman-dances-with-her-own-husband-1-if-a-pair-must-be-of-different-

Question Number 207787 by mr W last updated on 26/May/24 $$\boldsymbol{{n}}\:{married}\:{couples}\:{are}\:{invited}\:{to} \\ $$$${a}\:{dance}\:{party}.\:{for}\:{the}\:{first}\:{dance} \\ $$$$\boldsymbol{{n}}\:{paires}\:{are}\:{radomly}\:{selected}.\: \\ $$$${what}'{s}\:{the}\:{probability}\:{that}\:{no}\:{woman} \\ $$$${dances}\:{with}\:{her}\:{own}\:{husband}? \\ $$$$\left.\mathrm{1}\right)\:{if}\:{a}\:{pair}\:{must}\:{be}\:{of}\:{different} \\ $$$$\:\:\:\:\:{genders}. \\ $$$$\left.\mathrm{2}\right)\:{if}\:{a}\:{pair}\:{can}\:{also}\:{be}\:{of}\:{the}\:{same}\:…

Two-ships-have-the-same-berth-in-a-port-It-is-known-that-the-arrival-times-of-the-two-ships-are-independent-and-have-the-same-probability-of-docking-on-a-Sunday-00-00-24-00-If-the-berth-

Question Number 207752 by efronzo1 last updated on 25/May/24 $$\:\mathrm{Two}\:\mathrm{ships}\:\mathrm{have}\:\mathrm{the}\:\mathrm{same}\:\mathrm{berth}\: \\ $$$$\:\mathrm{in}\:\mathrm{a}\:\mathrm{port}.\:\mathrm{It}\:\mathrm{is}\:\mathrm{known}\:\mathrm{that}\:\mathrm{the}\: \\ $$$$\:\mathrm{arrival}\:\mathrm{times}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two}\:\mathrm{ships}\: \\ $$$$\:\mathrm{are}\:\mathrm{independent}\:\mathrm{and}\:\mathrm{have}\:\mathrm{the}\: \\ $$$$\:\mathrm{same}\:\mathrm{probability}\:\mathrm{of}\:\mathrm{docking}\: \\ $$$$\mathrm{on}\:\mathrm{a}\:\mathrm{Sunday}\:\left(\mathrm{00}.\mathrm{00}−\mathrm{24}.\mathrm{00}\right) \\ $$$$\:\mathrm{If}\:\mathrm{the}\:\mathrm{berth}\:\mathrm{time}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{ship} \\ $$$$\:\mathrm{is}\:\mathrm{2}\:\mathrm{hours}\:\mathrm{and}\:\mathrm{the}\:\mathrm{berth}\:\mathrm{time} \\…

It-is-known-that-a-balanced-6-sided-dice-originally-had-2-3-4-5-6-and-7-The-dice-wre-thrown-once-and-the-result-was-observed-If-an-odd-numbers-appears-than-the-number-is-replaced-with-the

Question Number 207751 by efronzo1 last updated on 25/May/24 $$\:\:\mathrm{It}\:\mathrm{is}\:\mathrm{known}\:\mathrm{that}\:\mathrm{a}\:\mathrm{balanced}\:\mathrm{6}−\mathrm{sided}\: \\ $$$$\:\mathrm{dice}\:\mathrm{originally}\:\mathrm{had}\:\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6}\:\mathrm{and}\:\mathrm{7}. \\ $$$$\:\mathrm{The}\:\mathrm{dice}\:\mathrm{wre}\:\mathrm{thrown}\:\mathrm{once}\:\mathrm{and}\: \\ $$$$\:\mathrm{the}\:\mathrm{result}\:\mathrm{was}\:\mathrm{observed}.\:\mathrm{If}\:\mathrm{an}\: \\ $$$$\mathrm{odd}\:\mathrm{numbers}\:\mathrm{appears},\:\mathrm{than}\:\mathrm{the}\: \\ $$$$\:\mathrm{number}\:\mathrm{is}\:\mathrm{replaced}\:\mathrm{with}\:\mathrm{the}\: \\ $$$$\:\mathrm{number}\:\mathrm{8}.\:\mathrm{However},\:\mathrm{if}\:\mathrm{an}\:\mathrm{even}\: \\ $$$$\:\mathrm{number}\:\mathrm{appears}\:,\:\mathrm{the}\:\mathrm{number} \\…

1-1-20-0-1-2-20-1-1-3-20-2-1-21-20-20-

Question Number 207665 by efronzo1 last updated on 22/May/24 $$\:\:\:\:\frac{\mathrm{1}}{\mathrm{1}}\begin{pmatrix}{\mathrm{20}}\\{\:\:\mathrm{0}}\end{pmatrix}\:+\frac{\mathrm{1}}{\mathrm{2}}\begin{pmatrix}{\mathrm{20}}\\{\:\:\mathrm{1}}\end{pmatrix}\:+\frac{\mathrm{1}}{\mathrm{3}}\begin{pmatrix}{\mathrm{20}}\\{\:\:\mathrm{2}}\end{pmatrix}\:+…+\frac{\mathrm{1}}{\mathrm{21}}\:\begin{pmatrix}{\mathrm{20}}\\{\mathrm{20}}\end{pmatrix}\:=? \\ $$ Answered by Tinku Tara last updated on 22/May/24 $$\left(\mathrm{1}+{x}\right)^{\mathrm{20}} =\underset{{n}=\mathrm{0}} {\overset{\mathrm{20}} {\sum}}\begin{pmatrix}{\mathrm{20}}\\{{n}}\end{pmatrix}{x}^{{n}} \\…

Let-cardE-n-and-the-set-of-parts-S-A-B-P-E-P-E-A-B-Show-that-cardS-3-n-

Question Number 207387 by sniper237 last updated on 13/May/24 $${Let}\:\:{cardE}={n}\:,\:{and}\:\:{the}\:{set}\:{of}\:{parts} \\ $$$${S}=\left\{\left({A},{B}\right)\in{P}\left({E}\right)×{P}\left({E}\right)\:/\:\:{A}\cap{B}=\varnothing\right\} \\ $$$${Show}\:{that}\:\:{cardS}=\:\mathrm{3}^{{n}} \\ $$ Answered by Berbere last updated on 13/May/24 $${if}\:{card}\left({A}\right)={k};{E}={A}\cup\overset{−} {{A}}\:\:{the}\:{number}\:{of}\:{subset}\:{of}\:{card}={k}…