A sample contains large number of nuclei. The probability that a nucleus in sample will decay after four half lives is

Explanation

Here we should remember the statement that after one half life, half of the nuclei decay is just an estimate. In reality, the decay of each nuclei is an independent process and we have to consider the probability that half of the nuclei decay during that time period and half of the nuclei survice. Note the word exactly.
Calculation : Out of the four nuclei available, we can choose any two nuclei that are to decay in
`""_(2)^(4)C=(4!)/(2!xx2!)=6` ways
The probability of survival of each nuclei in one half life is Ra.
Similary, the probability of decay of each nuclei in one half life is also
`""_(2)^(4)C(e^(-lamdat))2=3/8`
To examine it more deeply, we can say that in any time interval t, there can be two outcomes. Either the nuclei can decay or it can survive. so, the sum of the probability of survival and the probability of decay should be 1. In general, if n nuclei are being considered
`(P_(s)+P_(d))^(n)=1`
where P, respresents the probability of survival and `P_(d)`
represents the probability of decay.
The different terms in this binomial expression represent the probability of different outcomes. Thus, the probablity that all the nuclie will decay and none would survive will be
`""_(0)^(n)C(P_(2))^(0)(P_(d))^(n)`
and so on.