Find It Out !!!

Probability Level 5

A bag contains 7 black balls and an unknown number, not greater than seven , of white balls. The number of white balls range from 0 to 7, with equal probability.

Now, four balls are drawn successively without replacement and are all found to be white . What is the chance that a black ball will be drawn next ?

How To Answer

  • Answer as p + q p+q if your answer comes out to be of the form p q \frac{p}{q} , where g c d ( p , q ) = 1 gcd(p,q) = 1 .

Note

  • This problem is a part of the set 1+2=3


The answer is 6267.

Saurav Pal by Saurav Pal , 26, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Saurav Pal
Apr 9, 2015

You need to consider the following cases:

CASE 1 . 7 black balls and 4 white balls.
CASE 2 . 7 black balls and 5 white balls.
CASE 3 . 7 black balls and 6 white balls.
CASE 4 . 7 black balls and 7 white balls.


It is given in the question that four balls are drawn successively and all are found to be white, which implies that there are minimum 4 white balls present in the bag.

CASE 1 . 7 black balls and 4 white balls

Probability that all the 4 balls (which are drawn successively) are white = 4 11 . 3 10 . 2 9 . 1 8 = 1 330 \frac { 4 }{ 11 } .\frac { 3 }{ 10 } .\frac { 2 }{ 9 } .\frac { 1 }{ 8 } =\frac { 1 }{ 330 }

Probability that the next ball drawn is black = 1

CASE 2 . 7 black balls and 5 white balls

Probability that all the 4 balls (which are drawn successively) are white = 5 12 . 4 11 . 3 10 . 2 9 = 1 99 \frac { 5 }{ 12 } .\frac { 4 }{ 11 } .\frac { 3 }{ 10 } .\frac { 2 }{ 9 } =\frac { 1 }{ 99 }

Probability that the next ball drawn is black = 7 8 \frac{7}{8}

CASE 3 . 7 black balls and 6 white balls

Probability that all the 4 balls (which are drawn successively) are white = 6 13 . 5 12 . 4 11 . 3 10 = 3 143 \frac { 6 }{ 13 } .\frac { 5 }{ 12 } .\frac { 4 }{ 11 } .\frac { 3 }{ 10 } =\frac { 3 }{ 143 }

Probability that the next ball drawn is black = 7 9 \frac{7}{9}

CASE 4 . 7 black balls and 7 white balls

Probability that all the 4 balls (which are drawn successively) are white = 7 14 . 6 13 . 5 12 . 4 11 = 5 143 \frac { 7 }{ 14 } .\frac { 6 }{ 13 } .\frac { 5 }{ 12 } .\frac { 4 }{ 11 } =\frac { 5 }{ 143 }

Probability that the next ball drawn is black = 7 10 \frac{7}{10}

Now, the probability that a black ball will be drawn next is : ( 1. 1 330 + 7 8 . 1 99 + 7 9 . 3 143 + 7 10 . 5 143 ) ( 1 330 + 1 99 + 3 143 + 5 143 ) = 2711 3556 \frac { (1.\frac { 1 }{ 330 } +\frac { 7 }{ 8 } .\frac { 1 }{ 99 } +\frac { 7 }{ 9 } .\frac { 3 }{ 143 } +\frac { 7 }{ 10 } .\frac { 5 }{ 143 } ) }{ (\frac { 1 }{ 330 } +\frac { 1 }{ 99 } +\frac { 3 }{ 143 } +\frac { 5 }{ 143 } ) } = \boxed{\frac{2711}{3556}} .

∴ \therefore p + q = 6267 p+q=\boxed{6267} .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Please keep the solution discussions separate from the report forum. I have copied your comment over.

Note that you can edit your comments and solution, by clicking on the edit button.

Calvin Lin Staff - 6 years, 2 months ago

Log in to reply

Okay Sir. Thanks a lot.

Saurav Pal - 6 years, 2 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...