Math / Discrete

QuestionSlips of paper marked with the numbers $5,6,7$ , and 8 are placed in a box. After being mixed, two slips are drawn simultaneously. Write out the sample space $S$ , choosing an $S$ with equally likely outcomes, if possible. Then give the value of $n(S)$ and tell whether the outcomes in S are equally likely. Finally, write the indicated events below in set notation. a. Both slips are marked with even numbers. b. One slip is marked with an odd number and the other is marked with an even number. c. Both slips are marked with the same number.
What is the sample space? A. $S=\{(5,6),(5,8),(7,6),(7,8)\}$ B. $S=\{(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)\}$ C. $S=\{(5,7)\}$ D. $S=\{5,6,7,8\}$
The value of $n(S)$ is $\square$ .
Are the outcomes equally likely? No Yes a. Write the event both slips are marked with even numbers. Next

Studdy Solution

STEP 1

What is this asking? We're picking two slips of paper from a box with the numbers $5, 6, 7,$ and $8$ written on them, and we want to know all the possible pairs we can get, how many pairs there are, whether it's equally likely to get any pair, and then describe some specific scenarios. Watch out! Don't count the same pair twice (like picking $5$ then $6$ versus picking $6$ then $5$ )!
Also, make sure you understand what "equally likely" means – does every pair have the same chance of being picked?

STEP 2

1. List all possible pairs
2. Count the pairs
3. Check if outcomes are equally likely
4. Describe the specific events

STEP 3

Let's systematically list all the possible pairs of numbers we can draw.
We can start by considering all pairs involving $5$ : $(5,6)$ , $(5,7)$ , and $(5,8)$ .

STEP 4

Now, let's consider pairs involving $6$ , but remember we've already counted $(5,6)$ , so we only need to add $(6,7)$ and $(6,8)$ .

STEP 5

Finally, for pairs involving $7$ , we've already counted $(5,7)$ and $(6,7)$ , so we just need to add $(7,8)$ .
We've covered all the numbers, so our sample space is $S = \{(5,6), (5,7), (5,8), (6,7), (6,8), (7,8)\}$ .

STEP 6

We can count the number of pairs in our sample space $S$ .
There are six pairs, so $n(S) = 6$ .

STEP 7

Since we're drawing the slips simultaneously, each pair has an equal chance of being picked.
So, yes, the outcomes are equally likely.

STEP 8

The even numbers are $6$ and $8$ .
The only pair with both even numbers is $(6,8)$ .
So, the event is $\{(6,8)\}$ .

STEP 9

The odd numbers are $5$ and $7$ .
The even numbers are $6$ and $8$ .
The pairs with one odd and one even are $(5,6)$ , $(5,8)$ , $(7,6)$ , and $(7,8)$ .
So, the event is $\{(5,6), (5,8), (7,6), (7,8)\}$ .

STEP 10

This is impossible since we only draw two slips and each slip has a different number.
So, the event is an empty set, denoted by $\{\}$ or $\emptyset$ .

STEP 11

The sample space is $S = \{(5,6), (5,7), (5,8), (6,7), (6,8), (7,8)\}$ , which corresponds to option B. $n(S) = 6$ , and the outcomes are equally likely.
The event "both slips are even" is $\{(6,8)\}$ .
The event "one slip is odd, one is even" is $\{(5,6), (5,8), (7,6), (7,8)\}$ .
The event "both slips have the same number" is $\emptyset$ .

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Studdy Solution

STEP 1

STEP 2

1. List all possible pairs
2. Count the pairs
3. Check if outcomes are equally likely
4. Describe the specific events

STEP 3

Let's systematically list all the possible pairs of numbers we can draw.
We can start by considering all pairs involving $5$ : $(5,6)$ , $(5,7)$ , and $(5,8)$ .

STEP 4

Now, let's consider pairs involving $6$ , but remember we've already counted $(5,6)$ , so we only need to add $(6,7)$ and $(6,8)$ .

STEP 5

Finally, for pairs involving $7$ , we've already counted $(5,7)$ and $(6,7)$ , so we just need to add $(7,8)$ .
We've covered all the numbers, so our sample space is $S = \{(5,6), (5,7), (5,8), (6,7), (6,8), (7,8)\}$ .

STEP 6

We can count the number of pairs in our sample space $S$ .
There are six pairs, so $n(S) = 6$ .

STEP 7

Since we're drawing the slips simultaneously, each pair has an equal chance of being picked.
So, yes, the outcomes are equally likely.

STEP 8

The even numbers are $6$ and $8$ .
The only pair with both even numbers is $(6,8)$ .
So, the event is $\{(6,8)\}$ .

STEP 9

The odd numbers are $5$ and $7$ .
The even numbers are $6$ and $8$ .
The pairs with one odd and one even are $(5,6)$ , $(5,8)$ , $(7,6)$ , and $(7,8)$ .
So, the event is $\{(5,6), (5,8), (7,6), (7,8)\}$ .

STEP 10

This is impossible since we only draw two slips and each slip has a different number.
So, the event is an empty set, denoted by $\{\}$ or $\emptyset$ .

STEP 11

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Studdy Solution

STEP 1

STEP 2

1. List all possible pairs2. Count the pairs3. Check if outcomes are equally likely4. Describe the specific events

STEP 3

Let's **systematically list** all the possible pairs of numbers we can draw.We can start by considering all pairs involving 555: (5,6)(5,6)(5,6), (5,7)(5,7)(5,7), and (5,8)(5,8)(5,8).

STEP 4

Now, let's consider pairs involving 666, but remember we've already counted (5,6)(5,6)(5,6), so we only need to add (6,7)(6,7)(6,7) and (6,8)(6,8)(6,8).

STEP 5

STEP 6

We can **count** the number of pairs in our sample space SSS.There are **six** pairs, so n(S)=6n(S) = 6n(S)=6.

STEP 7

Since we're drawing the slips simultaneously, each pair has an equal chance of being picked.So, yes, the outcomes are **equally likely**.

STEP 8

The even numbers are 666 and 888.The only pair with both even numbers is (6,8)(6,8)(6,8).So, the event is {(6,8)}\{(6,8)\}{(6,8)}.

STEP 9

The odd numbers are 555 and 777.The even numbers are 666 and 888.The pairs with one odd and one even are (5,6)(5,6)(5,6), (5,8)(5,8)(5,8), (7,6)(7,6)(7,6), and (7,8)(7,8)(7,8).So, the event is {(5,6),(5,8),(7,6),(7,8)}\{(5,6), (5,8), (7,6), (7,8)\}{(5,6),(5,8),(7,6),(7,8)}.

STEP 10

This is impossible since we only draw two slips and each slip has a different number.So, the event is an **empty set**, denoted by {}\{\}{} or ∅\emptyset∅.

STEP 11

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1. List all possible pairs
2. Count the pairs
3. Check if outcomes are equally likely
4. Describe the specific events

Let's systematically list all the possible pairs of numbers we can draw.
We can start by considering all pairs involving $5$ : $(5,6)$ , $(5,7)$ , and $(5,8)$ .

Now, let's consider pairs involving $6$ , but remember we've already counted $(5,6)$ , so we only need to add $(6,7)$ and $(6,8)$ .

We can count the number of pairs in our sample space $S$ .
There are six pairs, so $n(S) = 6$ .

Since we're drawing the slips simultaneously, each pair has an equal chance of being picked.
So, yes, the outcomes are equally likely.

The even numbers are $6$ and $8$ .
The only pair with both even numbers is $(6,8)$ .
So, the event is $\{(6,8)\}$ .

The odd numbers are $5$ and $7$ .
The even numbers are $6$ and $8$ .
The pairs with one odd and one even are $(5,6)$ , $(5,8)$ , $(7,6)$ , and $(7,8)$ .
So, the event is $\{(5,6), (5,8), (7,6), (7,8)\}$ .

This is impossible since we only draw two slips and each slip has a different number.
So, the event is an empty set, denoted by $\{\}$ or $\emptyset$ .