A die is thrown 7 times. What is the chance that an odd number turns up (i) exactly 4 times (ii) at least 4 times

Explanation

Probability of success is 3/6=1/2.
`thereforep=1/2andq=1/2`
(i) For exactly 4 successes, the required probability is `""^(7)C_(4)((1)/(2))^(4)((1)/(2))^(3)=35/128`
(ii) For at least 4 successes, the required probability is `""^(7)C_(4)((1)/(4))^(4)((1)/(2))^(3)+""^(7)C_(5)((1)/(2))^(5)((1)/(2))^(2)`
`+""^(7)C_(6)((1)/(2))^(6)((1)/(2))^(1)+""^(7)C_(7)((1)/(2))^(7)`
`=35/128+21/128+7/128+1/128=64/128=1/2`