3. a jar contains 3 red marbles, 4 yellow marbles and 2 green marbles. you select 1 marble at random.
a. what is the probability the marble is yellow?
b. what is the probability the marble is either yellow or green?
c. what are the odds the marble is neither yellow nor green?
d. what are the odds of selecting a green marble?
4. on a science quiz, 6 students earned a b. there are 25 students in the class. if 2 students are chosen at random without replacement, what is the probability that both students earned a b?
5. how many different ways can you order 5 different library books on a shelf?
6. a single card is drawn from a standard deck with 52 cards. what is the probability that the card is a 5 or a heart? there are 13 cards in each suit, and 4 suits. what are the odds?
7. you roll a six sided die and flip a coin. what is the probability that you get an even number and a heads?
8. find the number of 3 letter words that can be formed using a 26 letter alphabet.
9. how many committees of 4 people can we choose from a group of 16 people?

Question

Accepted Answer

### Problem 3a
# Explanation:
## Step1: Find total number of marbles
Total marbles = 3 (red) + 4 (yellow) + 2 (green) = 9
## Step2: Calculate probability of yellow
Probability = $\frac{	ext{Number of yellow marbles}}{	ext{Total number of marbles}}$ = $\frac{4}{9}$
# Answer: $\frac{4}{9}$

### Problem 3b
# Explanation:
## Step1: Find number of yellow or green marbles
Yellow + Green = 4 + 2 = 6
## Step2: Calculate probability
Probability = $\frac{6}{9}$ = $\frac{2}{3}$ (simplified)
# Answer: $\frac{2}{3}$

### Problem 3c
# Explanation:
## Step1: Find number of marbles neither yellow nor green (red)
Red marbles = 3
## Step2: Find number of marbles that are yellow or green
Yellow + Green = 4 + 2 = 6
## Step3: Calculate odds
Odds = $\frac{	ext{Number of favorable}}{	ext{Number of unfavorable}}$ = $\frac{3}{6}$ = $\frac{1}{2}$ (odds in favor of neither yellow nor green)
# Answer: $\frac{1}{2}$ (or 1:2)

### Problem 3d
# Explanation:
## Step1: Number of green marbles (favorable)
Green marbles = 2
## Step2: Number of non - green marbles (unfavorable)
Non - green = 3+4 = 7
## Step3: Calculate odds
Odds = $\frac{2}{7}$ (odds in favor of green)
# Answer: $\frac{2}{7}$ (or 2:7)

### Problem 4
# Explanation:
## Step1: Probability of first student with B
Probability first = $\frac{6}{25}$
## Step2: Probability of second student with B (without replacement)
After one B student is chosen, there are 5 B students left and 24 total students. Probability second = $\frac{5}{24}$
## Step3: Multiply probabilities (independent events in sequence without replacement)
Total probability = $\frac{6}{25}	imes\frac{5}{24}$ = $\frac{30}{600}$ = $\frac{1}{20}$
# Answer: $\frac{1}{20}$

### Problem 5
# Explanation:
## Step1: Use permutation formula for arranging n distinct objects
The number of permutations of n distinct objects is $n!$ (n factorial). Here n = 5.
## Step2: Calculate 5!
5! = 5×4×3×2×1 = 120
# Answer: 120

### Problem 6 (Probability part)
# Explanation:
## Step1: Number of 5s in deck
There are 4 fives (one in each suit)
## Step2: Number of hearts
There are 13 hearts
## Step3: Number of cards that are 5 and heart (overlap)
There is 1 card (5 of hearts)
## Step4: Use inclusion - exclusion principle
Number of 5 or heart = 4+13 - 1 = 16
## Step5: Calculate probability
Probability = $\frac{16}{52}$ = $\frac{4}{13}$ (simplified)
# Answer: $\frac{4}{13}$

### Problem 7
# Explanation:
## Step1: Probability of even number on die
Even numbers on die (2,4,6) = 3. Probability = $\frac{3}{6}$ = $\frac{1}{2}$
## Step2: Probability of heads on coin
Probability of heads = $\frac{1}{2}$
## Step3: Multiply probabilities (independent events)
Total probability = $\frac{1}{2}	imes\frac{1}{2}$ = $\frac{1}{4}$
# Answer: $\frac{1}{4}$

### Problem 8
# Explanation:
## Step1: For each position in the 3 - letter word, we have 26 choices
Since the alphabet has 26 letters and we are forming 3 - letter words (with repetition allowed, as it's not specified otherwise), we use the multiplication principle.
## Step2: Calculate total number of words
Number of words = $26	imes26	imes26$ = $26^{3}$ = 17576
# Answer: 17576

### Problem 9
# Explanation:
## Step1: Use combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$
Here n = 16, r = 4
## Step2: Calculate factorials
$16! = 16	imes15	imes14	imes13	imes12!$, $4! = 4	imes3	imes2	imes1$, $(16 - 4)! = 12!$
## Step3: Simplify the combination formula
$C(16,4)=\frac{16!}{4!×12!}=\frac{16	imes15	imes14	imes13	imes12!}{4	imes3	imes2	imes1	imes12!}$ = $\frac{16	imes15	imes14	imes13}{24}$
## Step4: Calculate the value
16×15 = 240, 240×14 = 3360, 3360×13 = 43680; 43680÷24 = 1820
# Answer: 1820

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QUESTION IMAGE

Question

Explanation:

Problem 3a

Step1: Find total number of marbles

Step2: Calculate probability of yellow

Step1: Find number of yellow or green marbles

Step2: Calculate probability

Step1: Find number of marbles neither yellow nor green (red)

Step2: Find number of marbles that are yellow or green

Step3: Calculate odds

Answer:

Problem 3b