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 success? (iii) at least 6 success? (iv) at most 6 successes?

Answer

`(i) .((5)/(6))^(7) (ii).(35)/(6^(7)) (iii) .(1)/(6^(5)) (iv).(1-(1)/(6^(7)))`

Explanation

Let E ={(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}
`:. , p=(6)/(36) =(1)/(6),q=(1-(1)/(6))=(5)/(6),n=7`
`P(X=r)=.^(n)C_(r).p^(r ).q^((n-r)) =.^(7)C_(r ).((1)/(6))^(r ).((5)/(6))^((7-r))`
`(i) P(X=0) =.^(7)C_(0).((5)/(6))^(7)`
(ii) `P(X=6)=.^(7)C_( 6) .((1)/(6))^(6) .((5)/(6))`
(iii) P(at least 6 successes ) =P(6 successes or 7 successes )
`=P(X=6) +P(X=7)`
`=.^(7)C_(6).((1)/(6))^(6).((5)/(6)) +.^(7)C_(7).((1)/(6))^(7)`
(iv) P (at most 6 successes) =`P(X ge 6)`
`=[1 -P(X =7) ] = 1-.^(7)C_(7).((1)/(6))^(7)`