advanced math decision making
unit 2
2) you pick a day of the week, then choose a letter from the word math.
b) how many outcomes are possible?
c) what is the probability of picking day that starts with an s, then picking a t from the word math?
d) suppose you now had to pick a color, red or blue, after choosing your day of the week and letter, how many outcomes are possible now?
3) the following countries are written on pieces of paper and placed inside a hat: \cuba, jamaica, canada, america, mexico, haiti, grenada and honduras\. what is the probability of selecting a country that does not start with a c?
set theory
4) use the following venn diagram to answer a-d.
venn diagram with circles c and d, colors: c has blue, orange, green, red (overlap); d has purple, black, red (overlap), pink outside; yellow outside both
a) what colors are in d?
b) what colors are in only c?
c) what colors are in c ∩ d?
d) what colors are in c’ (the complement of c)?

Question

advanced math decision making
unit 2
2) you pick a day of the week, then choose a letter from the word math.
    b) how many outcomes are possible?
    c) what is the probability of picking day that starts with an s, then picking a t from the word math?
    d) suppose you now had to pick a color, red or blue, after choosing your day of the week and letter, how many outcomes are possible now?
3) the following countries are written on pieces of paper and placed inside a hat: \cuba, jamaica, canada, america, mexico, haiti, grenada and honduras\. what is the probability of selecting a country that does not start with a c?
set theory
4) use the following venn diagram to answer a-d.
venn diagram with circles c and d, colors: c has blue, orange, green, red (overlap); d has purple, black, red (overlap), pink outside; yellow outside both
    a) what colors are in d?
    b) what colors are in only c?
    c) what colors are in c ∩ d?
    d) what colors are in c’ (the complement of c)?

Accepted Answer

### 2b)
# Explanation:
## Step1: Count days of the week
There are 7 days in a week: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday. So, number of ways to pick a day is $ 7 $.
## Step2: Count letters in "MATH"
The word "MATH" has 4 letters: M, A, T, H. So, number of ways to pick a letter is $ 4 $.
## Step3: Use multiplication principle
To find the total number of outcomes, multiply the number of ways to pick a day by the number of ways to pick a letter. So, total outcomes $ = 7\times4 = 28 $.
# Answer:
$ 28 $

### 2c)
# Explanation:
## Step1: Count days starting with "S"
Days starting with "S" are Saturday and Sunday, so there are $ 2 $ such days.
## Step2: Count "T" in "MATH"
In the word "MATH", there is $ 1 $ "T".
## Step3: Calculate probability
Probability is the number of favorable outcomes divided by total outcomes (from 2b, total outcomes are $ 28 $). Favorable outcomes $ = 2\times1 = 2 $. So, probability $ = \frac{2}{28}=\frac{1}{14} $.
# Answer:
$ \frac{1}{14} $

### 2d)
# Explanation:
## Step1: Recall total outcomes from 2b
From 2b, we had $ 28 $ outcomes for day and letter.
## Step2: Count color choices
There are 2 color choices: red or blue.
## Step3: Use multiplication principle
Total new outcomes $ = 28\times2 = 56 $.
# Answer:
$ 56 $

### 3)
# Explanation:
## Step1: Count total countries
Countries: Cuba, Jamaica, Canada, America, Mexico, Haiti, Grenada, Honduras. Total of $ 8 $ countries.
## Step2: Count countries starting with "C"
Countries starting with "C": Cuba, Canada. So, $ 2 $ countries.
## Step3: Count countries not starting with "C"
Countries not starting with "C" $ = 8 - 2 = 6 $.
## Step4: Calculate probability
Probability $ = \frac{\text{Number of countries not starting with C}}{\text{Total number of countries}}=\frac{6}{8}=\frac{3}{4} $.
# Answer:
$ \frac{3}{4} $

### 4a)
# Brief Explanations:
To find colors in $ D $, we look at all regions of set $ D $ (the right circle and its intersection with $ C $). From the Venn diagram, the colors in $ D $ are Red (intersection), Purple, Black. Also, Pink is outside both but wait, no—wait, the universal set is the rectangle. Wait, the Venn diagram: set $ D $ is the right circle, so its regions are the part only in $ D $ (Purple, Black) and the intersection with $ C $ (Red). Wait, no, looking at the diagram: the right circle (D) has Red (intersection), Purple, Black. Wait, the rectangle has Yellow outside both, C has Blue, Orange, Green, Red (intersection), D has Red (intersection), Purple, Black, and Pink outside both? Wait, no, the diagram: left circle (C) has Blue, Orange, Green, Red (middle). Right circle (D) has Red (middle), Purple, Black. Outside both: Yellow, Pink. So colors in $ D $ are Red, Purple, Black.
# Answer:
Red, Purple, Black

### 4b)
# Brief Explanations:
"Only $ C $" means the part of set $ C $ that does not overlap with $ D $. From the Venn diagram, the colors in only $ C $ are Blue, Orange, Green.
# Answer:
Blue, Orange, Green

### 4c)
# Brief Explanations:
$ C\cap D $ is the intersection of sets $ C $ and $ D $, which is the region common to both circles. From the Venn diagram, the color in the intersection is Red.
# Answer:
Red

### 4d)
# Brief Explanations:
$ C' $ (complement of $ C $) is all elements not in $ C $. From the Venn diagram, elements not in $ C $ are the parts outside circle $ C $: Purple, Black (in $ D $ but not $ C $), Yellow, Pink (outside both circles).
# Answer:
Yellow, Pink, Purple, Black

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QUESTION IMAGE

Question

Explanation:

2b)

Step1: Count days of the week

Step2: Count letters in "MATH"

Step3: Use multiplication principle

Step1: Count days starting with "S"

Step2: Count "T" in "MATH"

Step3: Calculate probability

Step1: Recall total outcomes from 2b

Step2: Count color choices

Step3: Use multiplication principle

Answer:

2c)