Topic 2 of 7

Probability rules and conditional probability

Step 17 · Statistics and probability topics

Combine probabilities with the addition and complement rules, read conditional probabilities from two-way tables, and test whether events are independent.

Addition and complement rules

Probabilities run from 0, impossible, to 1, certain. The complement of A, everything that is not A, has probability 1 − P(A). For A or B, add the probabilities and take away the overlap so it is not counted twice.

P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

Example: Cards

Draw one card from a standard 52-card deck. What is the probability of a heart or a king?

There are 13 hearts and 4 kings, but the king of hearts is in both.
13 + 4 − 1 = 16 cards.

Answer $\frac{16}{52} = \frac{4}{13}$ , about 0.31

Example: At least one

Roll two dice. What is the probability of at least one six?

The complement is no sixes: 5 × 5 = 25 of the 36 outcomes.
1 − 25/36 leaves 11/36.

Answer $\frac{11}{36}$ , about 0.31

Conditional probability

P(A | B), read 'A given B', is the probability of A once you know B happened. Keep only the outcomes in B, and ask what fraction of them are also in A.

P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}

Group	Passed	Failed	Total
Studied	45	5	50
Did not study	15	35	50
Total	60	40	100

Example: From a two-way table

Using the table, find P(passed | studied) and P(studied | passed).

Of the 50 who studied, 45 passed: 45 ÷ 50 = 0.9.
Of the 60 who passed, 45 studied: 45 ÷ 60 = 0.75.

Answer0.9 and 0.75: the order matters

Example: The multiplication rule

A bag has 5 red and 3 blue marbles. Draw two without replacement. What is the probability both are red?

P(first red) = 5/8.
Given that, P(second red) = 4/7.
Multiply: 20/56.

Answer $\frac{5}{14}$ , about 0.36

Independence

A and B are independent when knowing that one happened does not change the chance of the other: P(A | B) = P(A). Then P(A and B) = P(A) × P(B).

Example: Checking independence

In the table, are studying and passing independent?

P(passed) = 0.6, but P(passed | studied) = 0.9.
Knowing that someone studied changes the chance they passed.

AnswerNo, they are not independent

Example: Independent trials

A free-throw shooter makes 80% of her shots, independently. What is the probability she makes 3 in a row?

Multiply the three chances: 0.8 × 0.8 × 0.8.

Answer0.512

Try it yourself

Work each one on paper first, then open the answer to check.

P(A) = 0.35. What is P(not A)?
Show answer
0.65
P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2. Find P(A or B).
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0.7
P(A and B) = 0.12 and P(B) = 0.3. Find P(A | B).
Show answer
0.4
Using the table above, find P(failed | did not study).
Show answer
0.7
P(A) = 0.5 and P(B) = 0.3, and A and B are independent. Find P(A and B).
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0.15
Toss three fair coins. What is the probability of at least one head?
Show answer
$\frac{7}{8}$