statistics - Using php to calculate odds for events to happen one or more times with different odds -

i got set of different events, own odds happen. there way calculate odds them combined, odds 0, 1 2 , on happen.

the math easy, number of calculations grow quick, hoping there function me.

example 3 factors :

event | yes | no     | 3%  | 97% b     | 4%  | 96% c     | 5%  | 95% 

0 happening : $a[no] * $ b[no] * $c[no]

1 happening : $a[yes] * $ b[no] * $c[no] + $a[no] * $ b[yes] * $c[no] + $a[no] * $ b[no] * $c[yes]

2 happening = $a[yes] * $ b[yes] * $c[no] + $a[yes] * $ b[no] * $c[yes] + $a[no] * $ b[yes] * $c[yes]

3 happening : $a[yes] * $ b[yes] * $c[yes]

this easy write in php, problem how scale if add more events. adding 1 more doubles number of caclulations, , code long.

so there easier way this? i'll grateful tips or ideas.

this quite slow implementation.

let's consider case 5 events.

odds of 0 events happening is:

$no[0]  * $no[1]  * $no[2]  * $no[3]  * $no[4] 

odds of 1 event happening is:

$no[0]  * $no[1]  * $no[2]  * $no[3]  * $yes[4] + $no[0]  * $no[1]  * $no[2]  * $yes[3] * $no[4]  + $no[0]  * $no[1]  * $yes[2] * $no[3]  * $no[4]  + ... 

where go through multiplications there 1 'yes' choice.

odds of 2 events happening is:

$no[0]  * $no[1]  * $no[2]  * $yes[3] * $yes[4] + $no[0]  * $no[1]  * $yes[2] * $no[3]  * $yes[4] + $no[0]  * $no[1]  * $yes[2] * $yes[3] * $no[4]  + ... 

where go through multiplications there 2 'yes' choices.

this can generalized: calculate odds of n events happening go through multiplications there n 'yes' choices.

now when need calculate odds 0 5 events happening, need go through possible combinations of yes/no choices , add each multiplication $odds[$yescount].

$no[0]  * $no[1]  * $no[2]  * $no[3]  * $no[4]   ; added $odds[0] $no[0]  * $no[1]  * $no[2]  * $no[3]  * $yes[4]  ; added $odds[1] $no[0]  * $no[1]  * $no[2]  * $yes[3] * $no[4]   ; added $odds[1] $no[0]  * $no[1]  * $no[2]  * $yes[3] * $yes[4]  ; added $odds[2] $no[0]  * $no[1]  * $yes[2] * $no[3]  * $no[4]   ; added $odds[1] ... $yes[0] * $yes[1] * $yes[2] * $yes[3] * $yes[4]  ; added $odds[5] 

there total of 2**5 = 32 different multiplications here, or 2**$eventcount.

it easy go through these cases if assign number each case 0 2**$eventcount-1, , use bits of number select whether 'yes' or 'no' choice of each event included in multiplication, , add each multiplication result $odds[$yescount]:

// number of events $eventcount = 5;  // odds of each event happening $yes = [ 0.10, 0.50, 0.32, 0.66, 0.99 ];  // odds of each event not happening $no = []; ($eventnumber = 0; $eventnumber < $eventcount; $eventnumber++) {   $no[$eventnumber] = 1 - $yes[$eventnumber]; }  // initialize combined $odds 0 $odds = []; ($n = 0; $n <= $eventcount; $n++) {   $odds[$n] = 0; }  // calculate combined odds ($case = 0; $case < 2 ** $eventcount; $case++) {   $multiplication = 1;   $yescount = 0;   ($eventnumber = 0; $eventnumber < $eventcount; $eventnumber++) {     if ($case & (1 << $eventnumber)) {       $yescount++;       $multiplication *= $yes[$eventnumber];     } else {       $multiplication *= $no[$eventnumber];     }   }   $odds[$yescount] += $multiplication; }  // show combined odds ($n = 0; $n <= $eventcount; $n++) {   echo "odds of " . $n . " events happening " . $odds[$n] . "<br>\n"; }