Exercise 28 Page 682

Two six-sided dice are rolled once. We are given a diagram that represents the events $A$ and $B.$

Each axis in this diagram represents the number rolled on one of the dice. Event $A$ is rolling a $2$ on at least one of the dice. We can see that $11$ of the $36$ possible outcomes are favorable outcomes for this event. Let's find the probability! To describe what event $B$ represents, let's add the numbers rolled on each dice.

Outcome	Sum
$(1,4)$	$1+4=5$
$(2,3)$	$2+3=5$
$(3,2)$	$3+2=5$
$(4,1)$	$4+1=5$

We can see that event $B$ is rolling numbers that add up to $5.$ We can see that $4$ of the $36$ possible outcomes are favorable outcomes for this event. Let's find the probability!

Are the Events Dependent or Independent?

Suppose event $A$ happens. The total possible outcomes then reduces from $36$ to $11.$ Additionally, the available favorable outcomes for $B$ reduces from $4$ to $2$ because only $2$ outcomes are favorable for both $A$ and $B.$ Let's find the probability of $B$ given $A.$ We can see that the occurrence of $A$ does affect the occurrence of $B.$ Therefore, these events are dependent.