Exam-style application: In a probability and random-variable setting, use sample space, conditional probability, independence, expected value, binomial model, and distribution interpretation. Define the sample space for flipping a fair coin three times and find the probability of getting exactly two heads. State the setup, compute carefully, and interpret the result.

Exam-style application: In a probability and random-variable setting, use sample space, conditional probability, independence, expected value, binomial model, and distribution interpretation. The heights of plants are normally distributed with $\mu=150$ cm and $\sigma=10$ cm. Interpret the z-score of a plant that is 135 cm tall. Provide a concise written explanation.

Exam-style application: In a probability and random-variable setting, use sample space, conditional probability, independence, expected value, binomial model, and distribution interpretation. Given $P(A)=0.4$, $P(B)=0.5$, and $P(A \cup B)=0.7$, determine if A and B are independent. State the setup, compute carefully, and interpret the result.

Exam-style application: In a probability and random-variable setting, use sample space, conditional probability, independence, expected value, binomial model, and distribution interpretation. In a class of 100 students, 60 study Spanish, 40 study French, and 20 study both. Find the conditional probability $P(\text{French} \mid \text{Spanish})$. Provide a concise written explanation.

Exam-style application: In a probability and random-variable setting, use sample space, conditional probability, independence, expected value, binomial model, and distribution interpretation. A game has outcomes of $X = \{0, 2, 5\}$ with probabilities 0.5, 0.3, and $0.2$ respectively. Calculate the expected value $E(X)$. State the setup, compute carefully, and interpret the result.

Exam-style application: In a probability and random-variable setting, use sample space, conditional probability, independence, expected value, binomial model, and distribution interpretation. For $X \sim \text{Binomial}(n=8, p=0.25)$, calculate the exact probability of exactly 2 successes. Provide a concise written explanation.