A biometric authentication system using fingerprints returns a value between $0$ and $1$ for each attempted match. The system was tested with $1000$ genuine fingerprints and $1000$ imposter fingerprints. The histogram shows the numbers of genuine and imposter attempts for different ranges.

Ex: $88$ of the $1000$ genuine fingerprints returned a value of n between $0\mathrm{.}4$ and $0\mathrm{.}5,$ while only $3$ imposter fingerprints returned a value in the same range.

Range of nImposter

fingerprintsGenuine

fingerprints$0$ n $<0\mathrm{.}164300\mathrm{.}1$ n $<0\mathrm{.}231910\mathrm{.}2$ n $<0\mathrm{.}329150\mathrm{.}3$ n $<0\mathrm{.}45490\mathrm{.}4$ n $<0\mathrm{.}53880\mathrm{.}5$ n $<0\mathrm{.}611330\mathrm{.}6$ n $<0\mathrm{.}701580\mathrm{.}7$ n $<0\mathrm{.}801760\mathrm{.}8$ n $<0\mathrm{.}901880\mathrm{.}9$ n $10192$

$($a$)$

Determine the threshold value of n such that less than $1\%$ of imposter attempts are accepted. How many false alarms are generated as a result?

$($b$)$

Determine the threshold value of n such that less than $1\%$ of genuine attempts are rejected. How many imposter attempts are accepted as a result?

$($c$)$

Determine the threshold value of n such that no genuine attempts are rejected. How many imposter attempts are accepted as a result?

Feedback?