Probability measures the likelihood of an event.
In business, probability is used for:
Example: Toss a coin. S = {H, T} Event E (Head) = {H}
If all outcomes are equally likely: P(E) = (Number of favourable outcomes) / (Total outcomes)
Properties:
For any two events A and B: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If A and B cannot happen together: P(A ∩ B) = 0 So P(A ∪ B) = P(A) + P(B)
Example: In a single roll of a die, event “2” and event “5” are mutually exclusive.
A and B are independent if occurrence of one does not affect the other. P(A ∩ B) = P(A) × P(B)
Example: Two coin tosses: P(H then H) = 1/2 × 1/2 = 1/4
If events are dependent: P(A ∩ B) = P(A) × P(B|A)
Example: Drawing two cards without replacement.
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Total balls = 3 + 2 = 5. Favourable outcomes (red) = 3. Probability = 3/5.
A = even {2,4,6} so P(A)=3/6=1/2. B = >4 {5,6} so P(B)=2/6=1/3. A∩B={6} so P(A∩B)=1/6. P(A∪B)=1/2 + 1/3 − 1/6 = 2/3.
Business mathematics are mathematics used by commercial enterprises to record and manage business operations. Commercial organizations use mathematics in accounting, inventory management, marketing, sales forecasting, and financial analysis.
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Probability measures the likelihood of an event.
In business, probability is used for:
Example: Toss a coin. S = {H, T} Event E (Head) = {H}
If all outcomes are equally likely: P(E) = (Number of favourable outcomes) / (Total outcomes)
Properties:
For any two events A and B: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If A and B cannot happen together: P(A ∩ B) = 0 So P(A ∪ B) = P(A) + P(B)
Example: In a single roll of a die, event “2” and event “5” are mutually exclusive.
A and B are independent if occurrence of one does not affect the other. P(A ∩ B) = P(A) × P(B)
Example: Two coin tosses: P(H then H) = 1/2 × 1/2 = 1/4
If events are dependent: P(A ∩ B) = P(A) × P(B|A)
Example: Drawing two cards without replacement.
Conditional probability of B given A: P(B|A) = P(A ∩ B) / P(A) (when P(A) > 0)
Bayes idea (basic): P(A|B) = [P(B|A) × P(A)] / P(B)
Expected value (EV) is the probability-weighted average of outcomes.
If outcomes are x1, x2, … with probabilities p1, p2, …: EV = Σ (pi × xi)
In decision making under risk, this is called Expected Monetary Value (EMV).
Decision rule (basic): Choose the alternative with higher EMV (risk-neutral decision).
List states of nature (e.g., High demand, Low demand) with probabilities and payoffs. Compute EMV for each decision.
Expected value at a chance node: EV = Σ (probability × payoff)
Expected value method mainly applies to risk.
A bag has 3 red and 2 blue balls. One ball is drawn. P(Red) = 3/(3+2) = 3/5.
From a standard die: A = even number (2,4,6) ⇒ P(A) = 3/6 = 1/2 B = number > 4 (5,6) ⇒ P(B) = 2/6 = 1/3 A ∩ B = {6} ⇒ P(A ∩ B) = 1/6 P(A ∪ B) = 1/2 + 1/3 − 1/6 = 2/3
A seller chooses Strategy 1 or Strategy 2. States of nature:
Payoffs (₹):
EMV(1) = 0.6×50,000 + 0.4×10,000 = 34,000 EMV(2) = 0.6×40,000 + 0.4×20,000 = 32,000
So Strategy 1 is better by EMV.
Tip: Write the formula first, then substitute values.
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Key probability rules:
Complement rule: P(A') = 1 − P(A)
Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) If A and B are mutually exclusive, P(A ∩ B)=0 so P(A ∪ B)=P(A)+P(B)
Multiplication rule: If A and B are independent: P(A ∩ B) = P(A)P(B) If dependent: P(A ∩ B) = P(A)P(B|A)
These rules are used to compute probabilities in business risk situations (e.g., joint events, alternative events).