Probability

Class 11 · Mathematics

Probability

Prev

Probability

Probability batati hai ki koi event hone ka kitna chance hai. Class 10 me humne classical probability dekhi; yahan ise zyada formal (axiomatic) tareeke se padhenge — sample space aur events ke through.

1. Basic Terms

  • Random experiment: aisa experiment jiska result pehle se pata nahi (jaise dice fekna).
  • Sample space \(S\): saare possible outcomes ka set. Dice ke liye \(S = \{1,2,3,4,5,6\}\).
  • Event: sample space ka koi subset. Jaise "even aana" \(= \{2,4,6\}\).
  • Outcome: sample space ka ek element.

2. Types of Events

  • Sure event: poora \(S\) (hamesha hota hai).
  • Impossible event: \(\varnothing\) (kabhi nahi hota).
  • Complementary event: \(A'\) = "\(A\) na ho".
  • Mutually exclusive: do events jo ek saath nahi ho sakte, yaani \(A \cap B = \varnothing\).
  • Exhaustive events: jinka union poora \(S\) ho.

3. Probability (Classical Definition)

Agar saare outcomes equally likely hon, to event \(A\) ki probability: \[ P(A) = \frac{\text{favourable outcomes ki ginti}}{\text{total outcomes ki ginti}} = \frac{n(A)}{n(S)}. \]

Har probability \(0 \le P(A) \le 1\) hoti hai. \(P(S) = 1\), \(P(\varnothing) = 0\).

Example

Ek fair dice me even number aane ki probability. \(A = \{2,4,6\}\), \(S = \{1,\dots,6\}\): \[ P(A) = \frac{3}{6} = \frac{1}{2}. \]

4. Addition Rule

Kisi do events ke liye: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B). \]

Agar \(A\) aur \(B\) mutually exclusive hain (\(A \cap B = \varnothing\)), to:

\[ P(A \cup B) = P(A) + P(B). \]

5. Complement Rule

\[ P(A') = 1 - P(A). \]

Aksar "kam se kam ek" type sawaal me, "na hone" ki probability nikaal kar \(1\) me se ghatana aasaan hota hai.

Example

52 cards me se ek card uthaya. King ya Heart aane ki probability? \[ P(\text{King}) = \frac{4}{52},\quad P(\text{Heart}) = \frac{13}{52},\quad P(\text{King} \cap \text{Heart}) = \frac{1}{52}. \] \[ P(\text{King} \cup \text{Heart}) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}. \]

Key Takeaways

  • Sample space \(S\) = saare outcomes; event = \(S\) ka subset.
  • \(P(A) = \dfrac{n(A)}{n(S)}\), aur \(0 \le P(A) \le 1\).
  • Addition: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\); mutually exclusive me \(P(A \cap B) = 0\).
  • Complement: \(P(A') = 1 - P(A)\) — "at least" type sawaalon me kaam aata hai.
Prev