NDA Probability: Events, Conditional Probability, Bayes and Binomial Distribution

Owned by UPSC NDA Mathematics syllabus. Exact NDA-relevant scope: Random experiments; outcomes and sample space; mutually exclusive and exhaustive, impossible and certain events; union, intersection and complement; classical and statistical probability; elementary probability theorems; conditional probability; Bayes theorem; random variables; binomial distribution and relevant experiments.

Random experiments; outcomes and sample space; mutually exclusive and exhaustive, impossible and certain events; union, intersection and complement; classical and statistical probability; elementary probability theorems; conditional probability; Bayes theorem; random variables; binomial distribution and relevant experiments.

Sets, fractions, combinations and algebra.

• Sample space
• Event
• Mutually exclusive events
• Independent events
• Conditional probability
• Random variable
• Binomial experiment

• P(A^c)=1-P(A)
• P(A union B)=P(A)+P(B)-P(A intersection B)
• P(A|B)=P(A intersection B)/P(B), P(B)>0
• If independent: P(A intersection B)=P(A)P(B)
• Bayes: P(A_i|B)=P(B|A_i)P(A_i)/sum_j P(B|A_j)P(A_j)
• Binomial: P(X=r)=C(n,r)p^r(1-p)^(n-r), r=0,...,n

• Classical equally-likely formula requires equally likely outcomes.
• Mutually exclusive is not the same as independent.
• Conditional probability denominator must be non-zero.
• Binomial model requires fixed number of trials, two outcomes per trial under the chosen success definition, constant p and independent trials.
• Bayes categories should form an appropriate partition for the standard form.

1. A fair die: P(even)=3/6=1/2.
2. If P(A)=0.6, P(B)=0.5 and P(A intersection B)=0.3, then P(A union B)=0.8. Since 0.3=0.6x0.5, A and B are independent in this numerical case.
3. For 4 independent trials with success p=0.25, probability of exactly 2 successes is C(4,2)(0.25)^2(0.75)^2.

• Adding probabilities of overlapping events without subtracting intersection.
• Treating mutually exclusive events as independent.
• Using nPr instead of nCr in binomial probability.
• Forgetting to check binomial assumptions.
• Reversing P(A|B) and P(B|A).

• Knowledge Gap: The concept, definition or theorem needed to solve the item is not understood.
• Recall Gap: The concept was learned but a formula, condition or relation cannot be retrieved accurately.
• Execution Error: The correct method is known but algebra, arithmetic, sign, substitution, diagram or step execution fails.
• Decision / Selection Error: The candidate chooses an inefficient method, continues a time-heavy path, or attempts a low-confidence item without adequate basis.
• Needs Review: Evidence is insufficient or contradictory; do not mislabel the error until another worked item is checked.
• Time pressure note: Time pressure is treated as a contributing condition, not automatically as the root cause.

Translate words into events before calculating. Draw a small set diagram, tree or table when conditional structure is unclear. Decide whether the task is counting, union / intersection, conditional probability, Bayes or binomial. Under negative marking, stop if the event model itself is uncertain rather than forcing arithmetic on a wrong model.

Negative marking is a real exam condition, but this page does not promise a universal attempt threshold. Use method certainty and option evidence before committing to a low-confidence answer.

• Required reviewer: NDA Mathematics faculty with demonstrable subject competence.