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Probability

Question
CBSEENMA12033912
Wired Faculty App

A pair of dice is thrown 7 times. If getting a total of 7 is considered a success, what is the probability of
(i) no success?    (ii) 6 successes?   (iii) at least 6 successes?    (iv) at most 6 successes?

Solution

Here n = 7
Let p be the probability of getting a total of 7.
therefore space space space space straight p space equals space 6 over 36 space equals 1 over 6 comma space space space straight q space equals space 1 minus 1 over 6 space equals 5 over 6
(i) P(no success) = P(0) = straight C presuperscript 7 subscript 0 space open parentheses 5 over 6 close parentheses to the power of 7 minus 0 end exponent space open parentheses 1 over 6 close parentheses to the power of 0 space equals space open parentheses 5 over 6 close parentheses to the power of 7
(ii) P(6 successes) = P(6) = straight C presuperscript 7 subscript 6 space open parentheses 5 over 6 close parentheses to the power of 7 minus 6 end exponent space open parentheses 1 over 6 close parentheses to the power of 6 space equals space 7 over 1 cross times 5 over 6 cross times open parentheses 1 over 6 close parentheses to the power of 6 space equals space 35 space open parentheses 1 over 6 close parentheses to the power of 7
(iii) P(at least 6 (successes) = P(6) + P(7)
                                            equals space straight C presuperscript 7 subscript 6 space open parentheses 5 over 6 close parentheses to the power of 7 minus 6 end exponent space open parentheses 1 over 6 close parentheses to the power of 6 space plus space straight C presuperscript 7 subscript 7 space open parentheses 5 over 6 close parentheses to the power of 7 minus 7 end exponent space open parentheses 1 over 6 close parentheses to the power of 7 equals space 7 space cross times space 5 over 6 space cross times space open parentheses 1 over 6 close parentheses to the power of 6 space plus space 1 space cross times space 1 space cross times space open parentheses 1 over 6 close parentheses to the power of 7 space equals space left parenthesis 35 plus 1 right parenthesis space open parentheses 1 over 6 close parentheses to the power of 7 equals space 36 space cross times space open parentheses 1 over 6 close parentheses to the power of 7 space equals space open parentheses 1 over 6 close parentheses to the power of 5
(iv) P(at most 6 successes) = P(not 7 success)
                                            equals space 1 minus straight P left parenthesis 7 right parenthesis space equals space 1 minus straight C presuperscript 7 subscript 7 space open parentheses 5 over 6 close parentheses to the power of 7 minus 7 end exponent space open parentheses 1 over 6 close parentheses to the power of 7 space equals space 1 minus open parentheses 1 over 6 close parentheses to the power of 7

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