Question

1. a basketball player shoots three free throws and has an 80% chance of making each shot. let x = the number of shots made
2. a student takes a 6 - question true - false quiz and guesses each question. let x = the number of questions answered correctly.
3. a baseball team plays a 4 - game series and has a 62% chance of winning any game. let x = the number of games won.
4. explain why binomial random variables are discrete, not continuous.

Explanation:

Step1: Recall binomial distribution criteria

A binomial experiment has fixed - number of trials ($n$), independent trials, two possible outcomes per trial (success/failure), and a constant probability of success ($p$).

Step2: Analyze problem 4

For the basketball player: $n = 3$ (three free - throws), probability of success $p=0.8$, each shot is independent, and there are two outcomes (make or miss).

Step3: Analyze problem 5

For the student taking the quiz: $n = 6$ (six questions), $p = 0.5$ (probability of guessing a true - false question correctly), questions are independent, and two outcomes (correct or incorrect).

Step4: Analyze problem 6

For the baseball team: $n = 4$ (four - game series), $p = 0.62$, games are independent, and two outcomes (win or lose).

Step5: Explain discreteness of binomial random variables

Binomial random variables represent the number of successes in a fixed number of trials. The number of successes can only take on non - negative integer values (e.g., 0, 1, 2, …, n). Since it can only take on a countable number of distinct values, it is discrete.

Answer:

1. $n = 3$, $p = 0.8$
2. $n = 6$, $p = 0.5$
3. $n = 4$, $p = 0.62$
4. Binomial random variables represent the number of successes in a fixed number of trials and can only take on non - negative integer values, so they are discrete.