Probability basics

Why data people need probability

Data is never perfectly clean or complete. A model says "this payment is 92% likely to be fraud". A survey of 1,000 people is used to guess what 10 crore people think. To read and build such numbers, you need the language of chance. That language is probability.

The words

• An experiment is something with an uncertain result, like tossing a coin or rolling a die.
• An outcome is one possible result: heads, or the number 4.
• The sample space is the list of all possible outcomes: {1, 2, 3, 4, 5, 6} for a die.
• An event is a group of outcomes you care about: "an even number" is {2, 4, 6}.
• P(A) is the probability of event A, a number from 0 (impossible) to 1 (certain). 0.25 is the same as 25%.

Counting equally likely outcomes

When every outcome is equally likely, P(event) = (outcomes in the event) ÷ (all outcomes). For a fair die, P(even) = 3 ÷ 6 = 0.5. For a card drawn from a full deck, P(ace) = 4 ÷ 52, about 0.077.

from fractions import Fraction

die = [1, 2, 3, 4, 5, 6]
even = [x for x in die if x % 2 == 0]
print(Fraction(len(even), len(die)))
print(len(even) / len(die))

Three rules that do most of the work

1. NOT (the complement): P(not A) = 1 − P(A). If the chance of rain is 0.3, the chance of no rain is 0.7.
2. AND for independent events: P(A and B) = P(A) × P(B). Two events are independent when one does not change the chance of the other, like two separate coin tosses. Two heads: 0.5 × 0.5 = 0.25.
3. OR: P(A or B) = P(A) + P(B) − P(A and B). We subtract the overlap so it is not counted twice. If the events cannot happen together, the overlap is 0 and you just add.

Probability from data

In real work we rarely have a perfect die. We count how often something happened in the past. If 12 of 4,000 UPI payments last week were fraud, the observed probability is 12 ÷ 4,000 = 0.003. This is called empirical probability. It is a good guess only if the past looks like the future and the sample is large enough.

Simulation: let the computer roll the dice

import random

random.seed(1)
rolls = [random.randint(1, 6) for _ in range(60000)]
print(round(rolls.count(6) / len(rolls), 3))

Common mistakes

• Adding probabilities of events that overlap. Fix: subtract P(A and B), or count the outcomes directly.
• Multiplying probabilities of events that are not independent (rain today and rain tomorrow are related). Fix: use conditional probability, the next topic.
• The gambler's fallacy: "the coin gave 5 heads, so tails is due". Each toss is independent; the chance is still 0.5.
• Getting a probability above 1 or below 0. That always means a mistake in counting.