directions: for the situations below, define a random variable x for the situation and then decide if they follow a binomial distribution model by commenting on the four requirements.
1) you roll a dnd dice (20 - sided), 20 times, and record the number that shows on the dice.
2) a basketball player can make 60% of their free throws. the coach plans on having a free throw shooting competition and the player will be shooting 100 shots.
3) from a standard deck of cards, you pull out a card, record the suit, put it back, and reshuffle. you continue until you get two spades in a row.
4) you are conducting a survey at your school to see how many students own smartphones. the probability that a student will own a smartphone is 0.85. you plan on having 200 students participate in your survey.

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Accepted Answer

# 1)
## Brief Explanations:
- **Define $X$**: Let $X$ be the number on the 20 - sided die in each roll.
- **Binomial Distribution Check**:
    - **Fixed number of trials ($n$)**: There are $n = 20$ trials (rolls).
    - **Independent trials**: Each roll of the die is independent of the others.
    - **Two outcomes**: A binomial distribution requires two outcomes (success/failure). Here, there are 20 possible outcomes (numbers 1 - 20), not two. So, it does not follow a binomial distribution.

# 2)
## Brief Explanations:
- **Define $X$**: Let $X$ be the number of free throws made.
- **Binomial Distribution Check**:
    - **Fixed number of trials ($n$)**: $n=100$ (100 free - throw attempts).
    - **Independent trials**: Each free - throw attempt is independent (assuming no fatigue or external factors affecting the player's performance between shots).
    - **Two outcomes**: Success (making the free throw) and failure (missing the free throw).
    - **Constant probability of success ($p$)**: $p = 0.6$ (the player's free - throw success rate). So, it follows a binomial distribution $X\sim B(n = 100,p=0.6)$.

# 3)
## Brief Explanations:
- **Define $X$**: Let $X$ be the number of times we draw a card until we get two spades in a row.
- **Binomial Distribution Check**:
    - **Fixed number of trials ($n$)**: The number of trials is not fixed. We stop when we get two spades in a row, and the number of trials can vary (it could be 2, 3, 4, etc.). So, it does not follow a binomial distribution.

# 4)
## Brief Explanations:
- **Define $X$**: Let $X$ be the number of students who own a smartphone.
- **Binomial Distribution Check**:
    - **Fixed number of trials ($n$)**: $n = 200$ (200 students surveyed).
    - **Independent trials**: Assuming each student's response is independent of the others.
    - **Two outcomes**: Success (student owns a smartphone) and failure (student does not own a smartphone).
    - **Constant probability of success ($p$)**: $p=0.85$. So, it follows a binomial distribution $X\sim B(n = 200,p = 0.85)$.

# Answer:
1) Does not follow a binomial distribution.
2) Follows a binomial distribution $X\sim B(100,0.6)$.
3) Does not follow a binomial distribution.
4) Follows a binomial distribution $X\sim B(200,0.85)$.

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