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

Monjur Alahi

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


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Salim Ahmed Kawsar

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Let p be its probability of success and q be the failure. Then p =2q. Also, p+q=1. It gives `p=2//3and q=1//3.`
P {4 successes in 6 trials} `=""^(6)C_(4)p^(4)q^(2)`
`=""^(6)C_(4)((2)/(3))^(4)((1)/(3))^(2)" "(1)`
P {5 successes in 6 trials} `=""^(6)C_(5)((2)/(3))^(5)((1)/(3))" "(2)`
P{6 successes in 6 trials} `=""^(6)C_(6)((2)/(3))^(6)((1)/(3))^(0)" "(3)`
Therefore, the probability that there are at least 4 successes is P {either 4 or 5 or 6 successes}
`=""^(6)C_(4)((2)/(3))^(4)((1)/(3))^(1)+""^(6)C_(5)((2)/(3))^(5)((1)/(3))+""^(6)C_(6)((2)/(3))^(6)=496/729`

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