Bite-size Stats

Probability Rules Cheat Sheet

Basic probability rules with examples 😎.

Complement rule

P(A) = 1 - P(A’)

• A’ (or Aᶜ) means “not A”

The probability that Anya will graduate high school is 0.9. What is the probability that she will not graduate?

P(not graduate) = 1 - P(graduate) = 1 - 0.9 = 0.1

Multiplication rules (joint probability)

P(A ∩ B) = P(A) * P(B|A) if A and B are dependent

• Two events are dependent if the outcome of the first affects the outcome of the second
• ∩ is the symbol for “intersection” (think of it as “and”: A and B)
• P(B|A) means “the probability of B happening given A has occurred”

If you draw two cards, without replacement, what is the probability that both cards are red?

P(red and red) = P(red) * P(red given red) = 26/52 * 25/51 = 0.245

P(A ∩ B) = P(A) * P(B) if A and B are independent

• Two events are independent if the outcome of the first does not affect the outcome of the second

If you draw two cards, with replacement, what is the probability that both cards are red?

P(red and red) = P(red) * P(red) = 26/52 * 26/52 = 0.25

P(A ∩ B) = 0 if A and B are mutually exclusive

• Two events are mutually exclusive (or disjoint) if they can’t happen at the same time

If you roll a 6-sided die, what is the probability of rolling a 2 and 6 at the same time?

P(2 and 6) = 0 because you can’t simultaneously roll two faces

Addition rules (union of events)

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

• ∪ is the symbol for “union” (think of it as “or”: A or B)

There are 14 boys and 10 girls in a math class. 5 boys and 6 girls got an A in the class. If a student is chosen at random, what is the probability of choosing a girl or an A student?

P(girl or A) = P(girl) + P(A) - P(girl and A) = 10/24 + 11/24 - 6/24 = 0.625

P(A ∪ B) = P(A) + P(B) if A and B are mutually exclusive

• Because P(A ∩ B) = 0 when A and B are mutually exclusive, you’re left with P(A) + P(B)

If you roll a 6-sided die, what is the probability of rolling a 2 or a 6?

P(2 or 6) = P(2) + P(6) = 1/6 + 1/6 = 0.333

Conditional probability

P(A|B) = P(A ∩ B) / P(B)

• P(B) ≠ 0

A math teacher gave her class two tests. 80% of the class passed the first test. 60% of the class passed both tests. What percent of those who passed the first test also passed the second test?

P(second given first) = P(first and second) / P(first) = 0.6/0.8 = 75%

P(A|B) = ( P(B|A) * P(A) ) / P(B)

• Known as Bayes’ Theorem

You planned to have a picnic today, but it’s cloudy this morning and 50% of all rainy days start off cloudy. However, cloudy mornings are common; about 40% of days start off cloudy. It’s also summer, and the chance of rain is only 8% this time of year. What is the probability of it raining today?