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 July 15th, 2011, 06:55 PM #1 Senior Member   Joined: Jul 2011 Posts: 405 Thanks: 16 probability A bag contains $5$ balls and it is not known that how many of these are white. $2$ balls are drawn and they are found to be white. What is the probability that all the balls in the bag were white?
 July 15th, 2011, 07:26 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,969 Thanks: 2219 Some additional information is needed to calculate this probability.
July 15th, 2011, 08:15 PM   #3
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Re: probability

Hello, panky!

Quote:
 A bag contains 5 balls and it is not known how many of them are white. Two balls are drawn and they are found to be white. What is the probability that all the balls in the bag were white?

Two balls are drawn from rive balls.
$\text{There are: }\:{5\choose2}\,=\,10\text{ possible outcomes.}$

$\text{Bayes' Formula: }\:P(\text{all W}\,|\,\text{2W drawn}) \;=\; \frac{P(\text{all W}\:\wedge\:\text{2W drawn})}{P(\text{2W drawn})}$[color=beige] .[/color][color=blue][A][/color]

Since two White balls were drawn, the bag contains at least two Whites.

There are four equally likely cases for the contents of the bag:
[color=beige]. . [/color][1] 2 Whites, 3 Blacks
[color=beige]. . [/color][2] 3 Whites, 2 Blacks
[color=beige]. . [/color][3] 4 Whites, 1 Black
[color=beige]. . [/color][5] 5 Whites
$\text{The probability of each case is }\frac{1}{4}$

[1] 2 Whites, 3 Blacks
$\text{There is: }\:{2\choose2} \,=\,1\text{ way to draw 2 Whites.}$
[color=beige]. . [/color]$P(\text{2W}) \:=\:\frac{1}{10}$
$P(\text{2W,\,3B}\:\wedge\:\text{2W drawn}) \:=\:\frac{1}{4}\,\cdot\,\frac{1}{10} \,=\,\frac{1}{40}$

[2] 3 Whites, 2 Blacks
$\text{There are: }\:{3\choose2} \,=\,3\text{ ways to draw 2 Whites.}$
[color=beige]. . [/color]$P(\text{2W}) \:=\:\frac{3}{10}$
$P(\text{3W,\,2B}\:\wedge\:\text{2W drawn}) \:=\:\frac{1}{4}\,\cdot\,\frac{3}{10} \,=\,\frac{3}{40}$

[3] 4 Whites, 1 Black
$\text{There are: }\:{4\choose2} \,=\,6\text{ ways to draw 2 Whites.}$
[color=beige]. . [/color]$P(\text{2W}) \:=\:\frac{6}{10}$
$P(\text{4W,\.1B}\:\wedge\:\text{2W drawn}) \:=\:\frac{1}{4}\,\cdot\,\frac{6}{10} \,=\,\frac{6}{40}$

[4] 5 Whites
$\text{There are: }\:{5\choose2} \,=\,10\text{ way to draw 2 Whites.}$
[color=beige]. . [/color]$P(\text{2W}) \:=\:\frac{10}{10}$
$P(\text{5W}\:\wedge\:\text{2W drawn}) \:=\:\frac{1}{4}\,\cdot\,\frac{10}{10} \,=\,\frac{10}{40}$[color=beige] .[/color][color=blue][B][/color]

$\text{Hence: }\:P(\text{2W drawn}) \;=\;\frac{1}{40}\,+\,\frac{3}{40}\,+\,\frac{6}{40 }\,+\,\frac{10}{40} \:=\:\frac{20}{40} \:=\:\frac{1}{2}$[color=beige] .[/color][color=blue][C][/color]

Substitute [color=blue][B][/color] and [color=blue][C][/color] into [color=blue][A}[/color]:
[color=beige]. . [/color]$P(\text{all W}\,|\,\text{2W drawn}) \;=\; \frac{P(\text{all W}\:\wedge\:\text{2W drawn})}{P(\text{2W drawn})} \;=\;\frac{\frac{10}{40}}{\frac{1}{2}} \;=\;\frac{1}{2}$

 July 18th, 2011, 07:05 PM #4 Senior Member   Joined: Jul 2011 Posts: 405 Thanks: 16 Re: probability thanks soroban

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# a bag contains 5 balls and it is not known how many of them white. two balls are drawn and these are found to be white. find probability that all the balls in the bag are white.

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