Question
Tue March 17, 2015 By: Pragyashree Jain

two balls are drawn from an urn containing 2 white 3 red and 4 black balls one by one without replacement. What is the probability that at least one ball is red?

Expert Reply
Vimala Ramamurthy
Thu March 19, 2015
L e t space A space b e space t h e space e v e n t space o f space g e t t i n g space a space n o n minus r e d space b a l l space i n space t h e space f i r s t space d r a w space a n d l e t space B space b e space t h e space e v e n t space o f space g e t t i n g space a space n o n minus r e d space b a l l space i n space t h e space s e c o n d space d r a w.  P open parentheses A close parentheses equals P r o b a b i l i t y space o f space g e t t i n g space a space n o n minus r e d space b a l l space i n space t h e space f i r s t space d r a w. rightwards double arrow P open parentheses A close parentheses equals 6 over 9 equals 2 over 3... left parenthesis 1 right parenthesis P open parentheses B close parentheses equals P r o b a b i l i t y space o f space g e t t i n g space a space n o n minus r e d space b a l l space i n space t h e space s e c o n d space d r a w. W h e n space a space n o n minus r e d space b a l l space i s space d r a w n space i n space t h e space f i r s t space d r a w comma space t h e r e space a r e space 5 space n o n minus r e d space b a l l s space a n d space 3 space r e d space b a l l s. T h e r e f o r e comma space f r o m space 5 space n o n minus r e d space b a l l s comma space o n e space b a l l space c a n space b e space s e l e c t e d space i n space C presuperscript 5 subscript 1 space w a y s. T h u s comma space P open parentheses B divided by A close parentheses equals 5 over 8.... left parenthesis 2 right parenthesis  R e q u i r e d space p r o b a b i l i t y equals P r o b a b i l i t y space t h a t space a t l e a s t space o n e space b a l l space i s space r e d space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 minus P r o b a b i l i t y space o f space g e t t i n g space n o n minus r e d space b a l l space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 minus P open parentheses A space a n d space B close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 minus P open parentheses A intersection B close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 minus P open parentheses A close parentheses cross times P open parentheses B divided by A close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 minus 2 over 3 cross times 5 over 8 space space space space space space space left square bracket f r o m space e q u a t i o n s space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis right square bracket space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 7 over 12
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