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Hi-A-ballot-box-contains-3-red-balls-4-blues-balls-and-5-white-balls-we-draw-successively-3-balls-in-ballot-box-by-re-puting-the-drawn-balls-1-Calculate-the-number-of-draws-containing-one-ball-o

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Question Number 89399 by mathocean1 last updated on 17/Apr/20
Hi. A ballot box contains 3 red balls, 4 blues balls and 5 white balls. we draw successively 3 balls in ballot box by re−puting the drawn balls. 1)Calculate the number of draws containing one ball of each color.
$$\mathrm{Hi}. \\ $$$$\mathrm{A}\:\mathrm{ballot}\:\mathrm{box}\:\mathrm{contains}\:\mathrm{3}\:\mathrm{red}\:\mathrm{balls},\:\mathrm{4}\:\mathrm{blues} \\ $$$$\mathrm{balls}\:\mathrm{and}\:\mathrm{5}\:\mathrm{white}\:\mathrm{balls}. \\ $$$$\mathrm{we}\:\mathrm{draw}\:\mathrm{successively}\:\mathrm{3}\:\mathrm{balls}\:\mathrm{in}\:\mathrm{ballot}\:\mathrm{box}\: \\ $$$$\mathrm{by}\:\mathrm{re}−\mathrm{puting}\:\mathrm{the}\:\mathrm{drawn}\:\mathrm{balls}. \\ $$$$\left.\mathrm{1}\right)\mathrm{Calculate}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{draws}\: \\ $$$$\mathrm{containing}\:\mathrm{one}\:\mathrm{ball}\:\mathrm{of}\:\mathrm{each}\:\mathrm{color}. \\ $$
Commented by mr W last updated on 17/Apr/20
the question is not clear at all to me! what means “by re−puting the drawn balls” exactly? does it mean that every time when a ball is drawn it will be put into the box before the next draw? or it means that at first three ball are drawn and then put into the box? what means “the number of draws containing one ball of each color”?
$${the}\:{question}\:{is}\:{not}\:{clear}\:{at}\:{all}\:{to}\:{me}! \\ $$$${what}\:{means}\:“{by}\:{re}−{puting}\:{the}\:{drawn} \\ $$$${balls}''\:{exactly}?\:{does}\:{it}\:{mean}\:{that}\:{every} \\ $$$${time}\:{when}\:{a}\:{ball}\:{is}\:{drawn}\:{it}\:{will}\:{be} \\ $$$${put}\:{into}\:{the}\:{box}\:{before}\:{the}\:{next}\:{draw}? \\ $$$${or}\:{it}\:{means}\:{that}\:{at}\:{first}\:{three}\:{ball}\:{are} \\ $$$${drawn}\:{and}\:{then}\:{put}\:{into}\:{the}\:{box}? \\ $$$${what}\:{means}\:“\mathrm{the}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{draws}\:\mathrm{containing}\:\mathrm{one}\:\mathrm{ball}\:\mathrm{of}\:\mathrm{each} \\ $$$$\mathrm{color}''? \\ $$
Commented by jagoll last updated on 17/Apr/20
3! C_1 ^3 ×C_1 ^4 ×C_1 ^5 = 6×3×4×5 = 18×20 = 360
$$\mathrm{3}!\:{C}_{\mathrm{1}} ^{\mathrm{3}} ×{C}_{\mathrm{1}} ^{\mathrm{4}} ×{C}_{\mathrm{1}} ^{\mathrm{5}} \:=\:\mathrm{6}×\mathrm{3}×\mathrm{4}×\mathrm{5} \\ $$$$=\:\mathrm{18}×\mathrm{20}\:=\:\mathrm{360} \\ $$
Commented by jagoll last updated on 17/Apr/20
according to my understand taking the ball one by one with return . the second ball is taken after the first ball is returned again
$${according}\:{to}\:{my}\:{understand}\: \\ $$$${taking}\:{the}\:{ball}\:{one}\:{by}\:{one}\:{with}\: \\ $$$${return}\:.\:{the}\:{second}\:{ball}\:{is}\:{taken}\: \\ $$$${after}\:{the}\:{first}\:{ball}\:{is}\:{returned}\: \\ $$$${again} \\ $$
Commented by mathocean1 last updated on 17/Apr/20
The first three balls are drawn and then put into the box...
$$\mathrm{The}\:\mathrm{first}\:\mathrm{three}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{drawn}\:\mathrm{and}\:\mathrm{then} \\ $$$$\mathrm{put}\:\mathrm{into}\:\mathrm{the}\:\mathrm{box}… \\ $$
Commented by mathocean1 last updated on 17/Apr/20
The draw is tricolor
$$\mathrm{The}\:\mathrm{draw}\:\mathrm{is}\:\mathrm{tricolor} \\ $$
Commented by mr W last updated on 17/Apr/20
then the question should be: three balls are drawn from the box, what is the probability that one ball of each color is drawn. right sir?
$${then}\:{the}\:{question}\:{should}\:{be}: \\ $$$${three}\:{balls}\:{are}\:{drawn}\:{from}\:{the}\:{box}, \\ $$$${what}\:{is}\:{the}\:{probability}\:{that}\:{one}\:{ball} \\ $$$${of}\:{each}\:{color}\:{is}\:{drawn}. \\ $$$${right}\:{sir}? \\ $$
Commented by mathocean1 last updated on 17/Apr/20
Yes sir... it′ s difficult for me to translate from french to english
$$\mathrm{Yes}\:\mathrm{sir}…\:\mathrm{it}'\:\mathrm{s}\:\mathrm{difficult}\:\mathrm{for}\:\mathrm{me}\:\mathrm{to}\:\mathrm{translate}\:\mathrm{from}\:\mathrm{french} \\ $$$$\mathrm{to}\:\mathrm{english}\: \\ $$
Commented by mr W last updated on 17/Apr/20
alright sir! now we know the exact question and can solve it.
$${alright}\:{sir}!\:{now}\:{we}\:{know}\:{the}\:{exact} \\ $$$${question}\:{and}\:{can}\:{solve}\:{it}. \\ $$
Commented by mr W last updated on 17/Apr/20
the box contains 12 balls. to draw three balls from the box there are C_3 ^(12) =220 ways. to draw three with three different colors there are 3×4×5=60 ways the probability is p=((60)/(220))=(3/(11))
$${the}\:{box}\:{contains}\:\mathrm{12}\:{balls}. \\ $$$${to}\:{draw}\:{three}\:{balls}\:{from}\:{the}\:{box}\:{there} \\ $$$${are}\:{C}_{\mathrm{3}} ^{\mathrm{12}} =\mathrm{220}\:{ways}. \\ $$$${to}\:{draw}\:{three}\:{with}\:{three}\:{different} \\ $$$${colors}\:{there}\:{are}\:\mathrm{3}×\mathrm{4}×\mathrm{5}=\mathrm{60}\:{ways} \\ $$$${the}\:{probability}\:{is} \\ $$$${p}=\frac{\mathrm{60}}{\mathrm{220}}=\frac{\mathrm{3}}{\mathrm{11}} \\ $$
Commented by jagoll last updated on 17/Apr/20
take the balls one by one instead of taking 3 at a time?
$${take}\:{the}\:{balls}\:{one}\:{by}\:{one}\: \\ $$$${instead}\:{of}\:{taking}\:\mathrm{3}\:{at}\:{a}\:{time}? \\ $$
Commented by jagoll last updated on 17/Apr/20
i think the answer is 6 × (3/(12))×(4/(12))×(5/(12)) = (5/(24))
$$\mathrm{i}\:\mathrm{think}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{is}\: \\ $$$$\mathrm{6}\:×\:\frac{\mathrm{3}}{\mathrm{12}}×\frac{\mathrm{4}}{\mathrm{12}}×\frac{\mathrm{5}}{\mathrm{12}}\:=\:\frac{\mathrm{5}}{\mathrm{24}} \\ $$$$ \\ $$

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