Math  /  Discrete

QuestionSuppose two dice (one red, one green) are rolled. Consider the following events. AA : the red die shows 1 ; BB : the numbers add to 3 ; CC : at least one of the numbers is 2 ; and DD : the numbers do not add to 9 . Express the given event in symbolic form. HINT [See Example 5.]
Either the numbers add to 9 or the red die shows a 1. DBD \cap B DAD \cap A DAD^{\prime} \cup A DAD^{\prime} \cap A DBD^{\prime} \cup B
How many elements does it contain? 9 \square

Studdy Solution

STEP 1

What is this asking? We need to figure out how to write "Either the numbers add to 9 *or* the red die shows a 1" using fancy math symbols, and then count how many dice rolls fit that description. Watch out! "Or" in math means "one, the other, or *both*"!
Don't forget that last part!

STEP 2

1. Translate to symbols
2. Count the possibilities

STEP 3

We're told that AA means "the red die shows 1".
Perfect! That's exactly one part of what we need.

STEP 4

We're also told that DD means "the numbers *do not* add to 9".
We want "the numbers *do* add to 9", which is the *opposite* of DD.
The opposite of an event is written with a little apostrophe, like this: DD'.

STEP 5

The problem says "Either...or...".
In math, "or" means "union", which we write with a \cup symbol.
So, "Either the numbers add to 9 or the red die shows a 1" translates to DAD' \cup A.

STEP 6

Let's **list out the ways** to get DD' (the numbers add to 9).
Remember, the dice have sides numbered 1 through 6.
We can have (3,6), (4,5), (5,4), and (6,3).
That's **4 ways**!

STEP 7

Now let's **list out the ways** to get AA (the red die shows 1).
We can have (1,1), (1,2), (1,3), (1,4), (1,5), and (1,6).
That's **6 ways**!

STEP 8

Now, remember what "or" means: one, the other, or *both*.
We need to add the number of ways for DD' and the number of ways for AA, but be careful!
Are there any overlaps?

STEP 9

Yes! The roll (1,6) is in both lists!
If we just add 4 + 6 = 10, we've counted (1,6) twice.
So, we need to subtract one to correct for the overlap.
That gives us 10 - 1 = **9 ways** total!

STEP 10

The symbolic form is DAD' \cup A, and there are **9** elements in the event.

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