Question

An urn contains 25 balls of which 10 balls bear a mark ‘X’ and the remaining 15 bear a mark ‘Y’ . A ball is drawn at random and it is replaced. If 6 balls are drawn in this way, find the probability that
(i) all will bear mark X
(ii) not more than 2 balls will bear 'Y' mark.
(iii) at least one ball will bear 'Y' mark.
(iv) the number of balls with 'X' mark and 'Y' mark will be equal.

Solution

Here n = 6
p = (Probability that a ball marked 'X' is drawn) = $10 over 25 space equals space 2 over 5$
$therefore space space space straight q space equals space straight P left parenthesis straight a space ball space marked space straight Y space is space drawn right parenthesis space equals space 1 space minus space straight p space equals space 1 minus 2 over 5 space equals space 3 over 5$
(i) P(all balls bear X mark) = P(6) = $straight C presuperscript 6 subscript 6 space open parentheses 3 over 5 close parentheses to the power of 6 minus 6 end exponent space open parentheses 2 over 5 close parentheses to the power of 6 space equals space open parentheses 2 over 5 close parentheses to the power of 6$
(ii) P(not more than 2 will bear Y mark)
= P(not less than 4 will bear X mark) = P(4) + P(5) + P(6)
$equals space straight C presuperscript 6 subscript 4 space open parentheses 3 over 5 close parentheses to the power of 6 minus 4 end exponent space open parentheses 2 over 5 close parentheses to the power of 4 space plus space straight C presuperscript 6 subscript 5 space open parentheses 2 over 5 close parentheses to the power of 6 minus 5 end exponent space plus space straight C presuperscript 6 subscript 6 space open parentheses 3 over 5 close parentheses to the power of 6 minus 6 end exponent space open parentheses 2 over 5 close parentheses to the power of 6 equals space straight C presuperscript 6 subscript 2 space open parentheses 3 over 5 close parentheses squared space open parentheses 2 over 5 close parentheses to the power of 4 space plus space straight C presuperscript 6 subscript 1 space open parentheses 3 over 5 close parentheses to the power of 1 space open parentheses 2 over 5 close parentheses to the power of 5 space plus space straight C presuperscript 6 subscript 0 space cross times space 1 space cross times space open parentheses 2 over 5 close parentheses to the power of 6 equals space fraction numerator 6 space cross times space 5 over denominator 1 space cross times 2 end fraction cross times space open parentheses 3 over 5 close parentheses squared space cross times space open parentheses 2 over 5 close parentheses to the power of 4 space plus space 6 over 1 cross times 3 over 5 cross times space open parentheses 2 over 5 close parentheses to the power of 5 space plus space 1 space cross times 1 space cross times space open parentheses 2 over 5 close parentheses to the power of 6 equals space open parentheses 2 over 5 close parentheses to the power of 4 space open square brackets fraction numerator 6 space cross times space 5 over denominator 1 space cross times 2 end fraction space cross times space 9 over 25 plus 6 over 1 cross times 3 over 5 cross times 2 over 5 plus 1 space cross times space 1 space cross times space 4 over 25 close square brackets space equals open parentheses 2 over 5 close parentheses to the power of 4 space open square brackets 27 over 5 plus 36 over 25 plus 4 over 25 close square brackets equals space open parentheses 2 over 5 close parentheses to the power of 4 space open square brackets fraction numerator 135 plus 36 plus 4 over denominator 25 end fraction close square brackets space equals space open parentheses 2 over 5 close parentheses to the power of 4 space cross times space 175 over 125 space equals space 7 open parentheses 2 over 5 close parentheses to the power of 4 space equals space 112 over 625$
(iii) P(number of balls with X mark and Y mark equal)
   $equals space straight P left parenthesis 3 right parenthesis space equals space straight C presuperscript 6 subscript 3 space open parentheses 3 over 5 close parentheses to the power of 6 minus 3 end exponent space open parentheses 2 over 5 close parentheses cubed space equals space fraction numerator 6 space cross times space 5 space cross times space 4 over denominator 1 space cross times space 2 space cross times space 3 end fraction cross times space open parentheses 3 over 5 close parentheses cubed space open parentheses 2 over 5 close parentheses cubed space equals space 20 space cross times space open parentheses 3 over 5 close parentheses cubed space cross times space open parentheses 2 over 5 close parentheses cubed$
(iv) P(at least one ball bear Y mark)
= P(not more than 5 balls bear mark X)
   $equals space 1 minus straight P left parenthesis 6 right parenthesis space equals space 1 space minus space open parentheses 2 over 5 close parentheses to the power of 6.$ $open square brackets because of space left parenthesis straight i right parenthesis close square brackets$

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