Formula sheet: Probability

Probability14 min readLesson 1 of 6

1.1 Events and sample spaces A random experiment has a set of possible outcomes, called the sample space S. An event is a subset of S: the event occurs when the outcome of the experiment lies in that subset. The whole sample space is the sure event and the empty set is the impossible event. Events can be combined with the usual set operations: A union B means at least one of A and B occurs, A intersection B means both occur, and the complement of A means A does not occur.

P (A \cup B) = P (A) + P (B) - P (A \cap B), P (A \cap B) = P (A) + P (B) - P (A \cup B)

Addition rule for two events and its rearrangement

1.2 Axioms of probability Probability assigns to every event a number between 0 and 1. The probability of the whole sample space is 1, and for mutually exclusive events the probabilities add. From these axioms the probability of the impossible event is 0, and the probability of not A is 1 minus the probability of A. Two events are mutually exclusive when they cannot occur together, so their intersection is empty and the addition rule drops the overlap term.

0 \leq P (A) \leq 1, P (S) = 1, P (ϕ) = 0, P (A^{'}) = 1 - P (A)

Basic consequences of the axioms of probability

P (A \cup B) = P (A) + P (B) (mutually exclusive), P (A \cup B \cup C) = P (A) + P (B) + P (C) - P (A \cap B) - P (B \cap C) - P (C \cap A) + P (A \cap B \cap C)

Addition rule for mutually exclusive events and for three events

1.3 Two useful consequences Two consequences are used in almost every exercise. First, the probability of not A is one minus the probability of A, which is handy when the complement is easier to count than the event itself. Second, if the events form a complete list of mutually exclusive possibilities, their probabilities add to one, so a missing probability can be found by subtracting the others from one. Both consequences come directly from the axioms and they often shorten a calculation considerably.

1.4 Equally likely outcomes When all outcomes are equally likely, the probability of an event is the number of favourable outcomes divided by the total number of outcomes. Counting the favourable outcomes is then a problem in permutations and combinations. For a die, a coin tossed several times, a pack of cards or a set of numbers, the sample space must be listed completely and without repetition before the ratio is formed.

P (E) = \frac{n ( E )}{n ( S )} = \frac{favourable outcomes}{total equally likely outcomes}

Classical probability for equally likely outcomes

Exam

Write the sample space first, then the event, then count. Most probability errors come from an incomplete sample space or from counting the same outcome twice.

in hinglish

Probability me pehle sample space likho, phir event ko subset ke roop me likho. Axioms yaad rakho: 0 se 1 ke beech probability, P(S) = 1, P(phi) = 0, aur P(A') = 1 - P(A). Agar do events saath nahi ho sakte to woh mutually exclusive hain aur P(A union B) = P(A) + P(B). Warna overlap subtract karna padta hai: P(A union B) = P(A) + P(B) - P(A intersection B). Equally likely outcomes me probability favourable outcomes by total outcomes hoti hai, isliye counting ka concept strong hona chahiye. Cards, coins aur dice ke standard sample spaces yaad rakho. Final answer ko 0 aur 1 ke beech check karo. Formula sheet ko revise karte waqt sample space size likhna aadat bana lo, isse denominator kabhi galat nahi aayega. Sample space aur event ko pehle likhne ki aadat ban jaye to probability ke sawaal aadhe solve ho jaate hain.