Probability And Statistics

Step 1: The total number of balls is . The probability of drawing a black ball on the first draw is: Step 2: After the first ball is drawn, the total number of balls left is 14. The probability of drawing a second black ball is: Step 3: The probability that both balls are black is the product of these two probabilities:

Step 1: Identify total Sundays in a leap year. A leap year has 366 days, which means 52 complete weeks + 2 extra days.

Step 2: Determine favorable cases. The extra 2 days can be: \begin{itemize} \item - (Sunday, Monday) \item - (Monday, Tuesday) \item - (Tuesday, Wednesday) \item - (Wednesday, Thursday) \item - (Thursday, Friday) \item - (Friday, Saturday) \item - (Saturday, Sunday) \end{itemize} In 2 out of 7 cases, Sunday is included.

Step 3: Compute probability.

Step 1: The events and are independent if:

Step 2: For a fair die, the probability of rolling an odd number (1, 3, or 5) is , so:

Step 3: The probability of both events occurring (i.e., rolling an odd number on both throws) is:

Step 4: Now check if : Since they are equal, the events and are independent.

Step 1: Define probabilities. Let be the probability that solves the problem. Let be the probability that solves the problem.

Step 2: Compute probability that neither solves it. Since the events are independent, Thus, the probability that none of them solve it is .

Step 3: Compute probability that at least one solves it. Using the complement rule: Thus, the probability that at least one of them solves it is .

Step 1: Using the formula for the probability of the complement: Step 2: Use the inclusion-exclusion principle to calculate : Substitute the given values: Step 3: Now calculate the complement: Thus, the answer is .

Step 1: Given probabilities. Let and .

Step 2: Find the probability that at least one wins. Using the formula for union of probabilities: Since nothing is mentioned about and winning together, we assume independence:

Step 3: Compute . Finding LCM of 3, 4, and 12:

Step 4: Find probability that neither wins. Thus, the probability that neither wins is \( \frac{1}{2} \

Step 1: We calculate , , and : - , so - , so - , so

Step 2: Check if the events are independent: For independent events, . Since , the events and are independent.

Step 1: The possible outcomes when a die is thrown twice and the sum is 6 are: So, there are 5 outcomes in total. Step 2: The favorable outcomes for the number 4 to appear at least once are: So, there are 2 favorable outcomes.

Step 3: The conditional probability is given by: Thus, the conditional probability is .