Independent and dependent events

Independent and dependent events, the multiplication rule, tree diagrams with and without replacement, and testing for independence.

Independent events

Events are independent if one does not change the probability of the other: $P(A \mid B) = P(A)$, which means $P(A \cap B) = P(A) \times P(B)$. Drawing with replacement gives independent events; without replacement gives dependent events.

Example 1●●

$P(A) = 0.4$, $P(B) = 0.5$ and $P(A \cap B) = 0.2$. Are $A$ and $B$ independent?

$P(A) \times P(B) = 0.2 = P(A \cap B)$

Yes, they are independent.

Dependent events on a tree

Example 2●●●

A bag has $5$ red and $3$ green balls. Two are drawn without replacement. Find $P(\text{same colour})$.

$P(RR) = \dfrac{5}{8} \times \dfrac{4}{7} = \dfrac{20}{56}$ and $P(GG) = \dfrac{3}{8} \times \dfrac{2}{7} = \dfrac{6}{56}$

$P = \dfrac{26}{56} = \dfrac{13}{28}$

Flashcards

Independent

$P(A \cap B) = P(A)P(B)$

Without replacement

Dependent events

Two aces without replacement

$\tfrac{4}{52} \times \tfrac{3}{51} = \tfrac{1}{221}$

Practice

$P(A) = 0.4$, $P(B) = 0.5$, $P(A \cap B) = 0.2$. Independent?
$5$ red, $3$ green, two without replacement. $P(\text{same colour})$?
Same bag. $P(\text{at least one green})$?
Two cards drawn without replacement. $P(\text{both aces})$?
$A$, $B$ independent, $P(A) = 0.3$, $P(B) = 0.5$. $P(A \cap B)$?
$P(A) = 0.3$, $P(B) = 0.6$, $P(A \cap B) = 0.1$. Independent?
Two cards drawn with replacement. $P(\text{both hearts})$?
$5$ red, $3$ green, two without replacement. $P(\text{red then green})$?