MathsHiranyagarbha · Class 11-12
Test positive aaya — ghabraane se pehle Bayes
Koi test 99% accurate ho aur tumhara result positive aaye — kya tumhe bimaari hone ka chance 99% hai?
Socho pehle… phir dekho jawab
Zaroori nahi! Agar bimaari bahut rare hai, toh zyaadatar positive results 'false positive' ho sakte hain. Jawab base rate (bimaari kitni aam hai) par depend karta hai — isi ka hisaab Bayes theorem deta hai.
Bayes: P(B | +) = P(+ | B) × P(B) / P(+). Yahan P(B) = base rate, aur P(+) = sabhi logon mein positive aane ka total chance (bimaar + swasth dono ke positive).
Ek example lo (numbers sirf samjhaane ke liye): 10,000 log, bimaari 1% mein, yaani 100 bimaar. Test bimaar ko 99% pakadta hai: 99 sahi positive. Swasth 9,900 mein se 1% galti se positive: 99 false positive. Kul positive 198, jinmein asli bimaar 99. Toh positive hone par bhi chance sirf 50%! Bimaari aur rare ho toh yeh aur gir jaata hai.
Trade-off: test ko zyada sensitive banao toh kam bimaar chhootenge, par false alarm badhenge; zyada strict banao toh false alarm kam, par kuch mareez chhoot jaayenge. Isliye doctor pehle screening, phir confirmatory test karwate hain. Yahi soch spam filter, fraud alert aur exam mein 'cheating detection' par bhi lagti hai.
Ghar pe try karo
Upar wale example mein bimaari ko 1% se 0.1% kar do (10,000 mein 10 bimaar) aur test same rakho. Table banao: sahi positive, false positive, kul positive. Positive hone par bimaari ka chance nikaalo — kitna gira?
Wow!
Rare cheezon ko dhoondhne wala 'accurate' test bhi zyaadatar galat alarm baja sakta hai — galti test ki nahi, base rate ki hoti hai.
3 sawaal
- Upar ke example mein positive result par bimaari ka chance kitna nikla?
Jawab dekho
50%. 198 positive mein 99 asli bimaar — yaani aadhe.
- Bayes theorem mein 'base rate' kya hai?
Jawab dekho
Population mein bimaari kitni aam hai, P(B). Prior probability P(B) ko base rate kehte hain.
- Bimaari aur rare ho jaaye (test same), toh positive par bimaari ka chance...
Jawab dekho
Ghat-ta hai. Swasth log zyada, toh false positive ka hissa badh jaata hai.
NCERT se jodo: NCERT Class 12 Maths — probability (conditional probability aur Bayes theorem)