1. Problem: In an experiment involving two successive rolls of a die, you are told that the sum of the two rolls is 9. How likely is it that the first roll was a 6? Step 1: List all possible pairs of rolls $(x,y)$ such that $x + y = 9$ where $x$ and $y$ are integers from 1 to 6. Step 2: The pairs are $(3,6), (4,5), (5,4), (6,3)$. Step 3: There are 4 possible pairs in total. Step 4: The pairs where the first roll is 6 is only $(6,3)$. Step 5: Probability that the first roll was 6 given the sum is 9 is $$\frac{1}{4} = 0.25.$$ 2. Problem: In a word guessing game, the first letter of the word is a “t”. What is the likelihood that the second letter is an “h”? Step 1: This depends on the dictionary or word list used. Assuming English words and that the first letter is "t", we consider the conditional probability $P(\text{second letter} = h \mid \text{first letter} = t)$. Step 2: Common English words starting with "th" are frequent (e.g., the, this, that). Step 3: Without exact data, we cannot compute a precise probability, but it is known that "th" is a common bigram in English. Step 4: If you have a corpus, count words starting with "th" divided by words starting with "t" to get the probability. 3. Problem: How likely is it that a person has a disease given that a medical test was negative? Step 1: Let $D$ be the event "person has disease" and $N$ be the event "test is negative". Step 2: We want $P(D \mid N)$. Step 3: By Bayes' theorem, $$P(D \mid N) = \frac{P(N \mid D) P(D)}{P(N)}.$$ Step 4: $P(N \mid D)$ is the false negative rate, $P(D)$ is the prevalence of the disease, and $P(N)$ is the total probability of a negative test. Step 5: $P(N) = P(N \mid D)P(D) + P(N \mid D^c)P(D^c)$ where $D^c$ is no disease. Step 6: Substitute values to compute $P(D \mid N)$. 4. Problem: A spot shows up on a radar screen. How likely is it that it corresponds to an aircraft? Step 1: Let $A$ be the event "spot is an aircraft" and $S$ be the event "spot shows up". Step 2: We want $P(A \mid S)$. Step 3: By Bayes' theorem, $$P(A \mid S) = \frac{P(S \mid A) P(A)}{P(S)}.$$ Step 4: $P(S \mid A)$ is the probability the radar detects an aircraft (true positive rate), $P(A)$ is the prior probability of an aircraft being present. Step 5: $P(S) = P(S \mid A)P(A) + P(S \mid A^c)P(A^c)$ where $A^c$ is no aircraft (false positive rate). Step 6: Substitute values to find $P(A \mid S)$.