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Probability

Question
CBSEENMA12033916
Wired Faculty App

If a fair coin is tossed 10 times, find the probability of
(i) exactly six heads
(ii) at least six heads
(iii) at most six heads

Solution

Here n = 10
P = Probability of head = 1 half comma space space straight q space equals space 1 space minus space straight p space equals space 1 minus 1 half space equals space 1 half
(i) P(exactly six heads) = P(6)
                 equals space straight C presuperscript 10 subscript 6 space open parentheses 1 half close parentheses to the power of 4 space open parentheses 1 half close parentheses to the power of 6 space equals space straight C presuperscript 10 subscript 4 space cross times space open parentheses 1 half close parentheses to the power of 10 equals space fraction numerator 10 space cross times space 9 space cross times space 8 space cross times space 7 over denominator 1 space cross times space 2 space cross times 3 space cross times 4 end fraction cross times 1 over 1024 space equals 105 over 512
(ii) P(at least six heads) = P(6) + P(7) + P(8) + P(9) + P(10)
             equals space straight C presuperscript 10 subscript 6 space open parentheses 1 half close parentheses to the power of 4 space open parentheses 1 half close parentheses to the power of 6 space plus space straight C presuperscript 10 subscript 7 space space open parentheses 1 half close parentheses cubed space open parentheses 1 half close parentheses to the power of 7 space plus space straight C presuperscript 10 subscript 8 space open parentheses 1 half close parentheses squared space open parentheses 1 half close parentheses to the power of 8 space space space space space space space space space space space space space space space space space space
             equals space open parentheses 1 half close parentheses to the power of 10 space open square brackets straight C presuperscript 10 subscript 6 space plus space straight C presuperscript 10 subscript 7 space plus space straight C presuperscript 10 subscript 8 space plus space straight C presuperscript 10 subscript 9 space plus space straight C presuperscript 10 subscript 10 close square brackets equals space open parentheses 1 half close parentheses to the power of 10 space open square brackets straight C presuperscript 10 subscript 4 space plus space straight C presuperscript 10 subscript 3 space plus space straight C presuperscript 10 subscript 2 space plus space straight C presuperscript 10 subscript 1 space plus space straight C presuperscript 10 subscript 0 space close square brackets space equals space open parentheses 1 half close parentheses to the power of 10 space open square brackets fraction numerator 10 space cross times space 9 space cross times space 8 space cross times 7 over denominator 1 space cross times 2 space cross times 3 space cross times 4 end fraction plus fraction numerator 10 space cross times space 9 space cross times 8 over denominator 1 space cross times 2 space cross times 3 end fraction plus fraction numerator 10 space cross times space 9 over denominator 1 space cross times 2 end fraction plus 10 over 1 plus 1 close square brackets equals space 1 over 1024 left square bracket 210 plus 120 plus 45 plus 10 plus 1 right square bracket space equals space 1 over 1024 space cross times space 386 space equals space 193 over 512.  
                                                       plus space straight C presuperscript 10 subscript 9 space open parentheses 1 half close parentheses space open parentheses 1 half close parentheses to the power of 9 space plus space straight C presuperscript 10 subscript 10 space open parentheses 1 half close parentheses to the power of 10
(iii) P(at most sixes) = P(0) + P(1) + P(2) + P(3) + P(4) + P(5) + P(6)
          equals space straight C presuperscript 10 subscript 0 space open parentheses 1 half close parentheses to the power of 10 space plus space straight C presuperscript 10 subscript 1 space open parentheses 1 half close parentheses to the power of 9 space open parentheses 1 half close parentheses space plus straight C presuperscript 10 subscript 2 space open parentheses 1 half close parentheses to the power of 8 space open parentheses 1 half close parentheses squared plus straight C presuperscript 10 subscript 3 space open parentheses 1 half close parentheses to the power of 7 space open parentheses 1 half close parentheses cubed
                                       plus space straight C presuperscript 10 subscript 4 space open parentheses 1 half close parentheses to the power of 6 space open parentheses 1 half close parentheses to the power of 4 space plus space straight C presuperscript 10 subscript 5 space open parentheses 1 half close parentheses to the power of 5 space open parentheses 1 half close parentheses to the power of 5 space plus space straight C presuperscript 10 subscript 4 space open parentheses 1 half close parentheses to the power of 6 space open parentheses 1 half close parentheses to the power of 4
equals space open parentheses 1 half close parentheses to the power of 10 space open square brackets straight C presuperscript 10 subscript 0 space plus space straight C presuperscript 10 subscript 1 space plus space straight C presuperscript 10 subscript 2 space plus space straight C presuperscript 10 subscript 3 space plus space straight C presuperscript 10 subscript 4 space plus space straight C presuperscript 10 subscript 5 space plus space straight C presuperscript 10 subscript 4 close square brackets equals space open parentheses 1 half close parentheses to the power of 10 space open square brackets 1 plus 10 over 1 plus fraction numerator 10 space cross times space 9 over denominator 1 space cross times 2 end fraction plus space fraction numerator 10 cross times 9 cross times space 8 over denominator 1 space cross times 2 space cross times 3 end fraction space plus fraction numerator 10 space cross times 9 cross times 8 cross times 7 over denominator 1 space cross times space 2 space cross times 3 space cross times 4 end fraction plus fraction numerator 10 space cross times space 9 space cross times 8 space cross times space 7 space cross times 6 over denominator 1 space cross times space 2 space cross times space 3 space cross times 4 cross times 5 end fraction plus fraction numerator 10 space cross times 9 cross times 8 cross times 7 over denominator 1 cross times 2 cross times 3 cross times 4 end fraction close square brackets equals space 1 over 1024 left square bracket 1 plus 10 plus 45 plus 120 plus 210 plus 252 plus 120 right square bracket space equals 848 over 1024 space equals 53 over 64.

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