Statistics

Independent Events

Lesson

Two events are independentwhen the outcome of one doesn’t affect the other. Flipping a coin and rolling a die. Drawing a marble, putting it back, drawing another. Two free throws in the same game (assuming the player’s skill is fixed).

For independent events, the probability they both happen is the product of their individual probabilities — the multiplication rule:

P(A \text{ and } B) = P(A) \cdot P(B)

It extends to as many independent events as you like — just multiply them all together.

A common technique: “at least one” problems are easier with the complement. To find $P(\text{at least one})$ , find $P(\text{none})$ first and subtract from 1.

Worked example 1

What’s the probability of rolling a 6 on a die and getting heads on a coin flip?

P(6 \text{ and heads}) = \tfrac{1}{6} \cdot \tfrac{1}{2} = \tfrac{1}{12}

Worked example 2 — “at least one”

Roll a die twice. What’s the probability of getting at least one 6?

Easier to count the complement: no 6 either time. $P(\text{not 6}) = 5/6$ each roll, and the rolls are independent:

\begin{aligned} P(\text{no 6 either time}) &= \tfrac{5}{6} \cdot \tfrac{5}{6} \\ &= \tfrac{25}{36} \end{aligned}

P(\text{at least one 6}) = 1 - \tfrac{25}{36} = \tfrac{11}{36}

How to type your answer

Fraction in lowest terms or a decimal. Examples: 1/4, 1/36, 11/36, 0.16.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

P(\text{heads then heads on two fair coins})

Problem 2

P(\text{rolling a 6, then a 6, on two fair dice})

Problem 3

\text{4 red, 6 blue marbles. Draw, replace, draw. } P(\text{red twice})

Problem 4

P(\text{heads three times in a row})

Practice

Standard problems matching the lesson.

Problem 5

P(\text{rolling even, then odd, on two dice})

Problem 6

\text{Draw, replace, draw. } P(\text{heart then king})

Problem 7

P(\text{6 on a die AND heads on a coin})

Problem 8

P(\text{first die shows 3, second die shows 5})

Problem 9

P(\text{HHT in that specific order})

Problem 10

\text{Spinner with 1/4 chance of red. } P(\text{red twice in a row})

Problem 11

P(\text{rolling NOT a 6, twice in a row})

Problem 12

\text{2\% defective rate. } P(\text{both of two parts defective})

Problem 13

P(\text{rolling doubles on two dice})

Problem 14

\text{Draw, replace, draw. } P(\text{ace then king})

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

P(\text{5 coin flips all heads})

Problem 16

\text{70\% free-throw shooter. } P(\text{makes 3 in a row})

Problem 17

P(\text{at least one head in 3 flips})

Problem 18

P(\text{4 dice all showing 6})

Problem 19

P(\text{at least one 6 in two rolls})

Problem 20

\text{10 marbles, 3 red. Draw with replacement, 3 times. } P(\text{all three red})

Problem 21

\text{Light is green 40\% of the time. } P(\text{green twice})

Problem 22

\text{1/3 chance of winning per spin. } P(\text{win 4 in a row})

Practice

Standard problems matching the lesson.

Problem 23

Flip a fair coin and roll a fair die. P(heads AND a 6)?

Problem 24

It rains on 30% of days. You forget your umbrella on 20% of days, independent of rain. P(both happen on the same day)?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

A basketball player makes each free throw with probability 0.8, independently. P(makes all 3 of 3 attempts)?

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