Question

An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes.

Solution

Here n = 6
Now, p + q = 1 and p = 2q
$therefore space space space 2 space straight q plus straight q space equals space 1 space space space rightwards double arrow space space space space straight q space equals space 1 third comma space space space therefore space space space space straight p space equals space 2 over 3.$
$therefore$ P(at least 4 successes in 6 trials) = $straight P left parenthesis straight X greater or equal than 4 right parenthesis space equals space straight P left parenthesis 4 right parenthesis space plus space straight P left parenthesis 5 right parenthesis space plus space straight P left parenthesis 6 right parenthesis$
$equals space straight C presuperscript 6 subscript 4 space straight q squared space straight p to the power of 4 plus space straight C presuperscript 6 subscript 5 space straight q space straight p to the power of 5 space plus space straight C presuperscript 6 subscript 6 space straight p to the power of 6 equals space straight p to the power of 4 space open square brackets straight C presuperscript 6 subscript 2 space end subscript straight q squared space plus space straight C presuperscript 6 subscript 1 space straight p space straight q space plus space straight C presuperscript 6 subscript 0 space straight p squared close square brackets equals space open parentheses 2 over 3 close parentheses to the power of 4 space open curly brackets fraction numerator 6 cross times 5 over denominator 1 cross times 2 end fraction open parentheses 1 third close parentheses squared space plus space 6 over 1.2 over 3.1 third plus 1 space cross times space open parentheses 2 over 3 close parentheses squared close curly brackets equals space 16 over 81 open curly brackets 15 over 9 plus 12 over 9 plus 4 over 9 close curly brackets space equals space 16 over 81 cross times 31 over 9 space equals 496 over 729.$

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Probability

An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes.

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