Probability basics and sampling: the rules behind every test question

Probability questions in aptitude and analyst screenings look varied but use four rules. Sampling questions ask whether a conclusion from a sample is trustworthy. Both reward slow, written-out working.

The four rules

• A probability is between 0 and 1. The probabilities of all possible outcomes add to 1.
• Complement: P(not A) = 1 − P(A).
• AND for independent events: P(A and B) = P(A) × P(B). Independent means one does not change the other.
• OR: P(A or B) = P(A) + P(B) − P(A and B). If A and B cannot both happen, the last term is 0.

Example 1: a fair six-sided die. P(even) = 3/6 = 0.5. P(greater than 4) = 2/6. P(even or greater than 4): even is {2, 4, 6}, greater than 4 is {5, 6}, both is {6}. So 3/6 + 2/6 − 1/6 = 4/6 = 2/3. Check by listing: {2, 4, 5, 6} is four of six.

Example 2: two independent machines each fail on a given day with probability 0.1. P(both fail) = 0.1 × 0.1 = 0.01. P(at least one fails) = 1 − P(neither fails) = 1 − 0.9 × 0.9 = 1 − 0.81 = 0.19.

Conditional probability from a table

P(A given B) means: among the cases where B is true, what fraction also have A? From a table, it is the cell count divided by the row or column total for B, not the grand total.

Example 3: 200 customers. 120 use the app and 80 use the website. Of the app users, 90 made a repeat purchase; of the website users, 40 did. P(repeat) = (90 + 40)/200 = 130/200 = 0.65. P(repeat given app) = 90/120 = 0.75. P(app given repeat) = 90/130, about 0.69. The last two differ: "given app" divides by app users, "given repeat" divides by repeat buyers. Swapping them is the classic error.

Sampling and bias

A sample is useful only if every member of the population had a fair chance of being included. A simple random sample picks at random from a full list. A convenience sample takes whoever is easy to reach.

Example 4: a company emails a satisfaction survey and a small share reply, mostly people with complaints. The average score is low, but it describes the people who chose to reply, not all customers. This is non-response bias, and a larger sample does not fix it.

Common mistakes and a practice habit

• Adding probabilities for OR without subtracting the overlap.
• Multiplying for AND when the events are not independent.
• Dividing by the grand total in a conditional probability.

Make a two-by-two table from anything you track, such as emails opened by day of week. Compute the four conditional probabilities, writing each fraction before converting it. Five minutes a day makes the "given" direction automatic.