You are taking a timed online quantitative assessment (in the style of Optiver's "Beat The Odds" test). It consists of rapid-fire probability questions that must be answered quickly with mental math — no calculator. For each question below, give the exact probability (as a fraction or decimal) and be ready to justify it in one or two sentences. ### Constraints & Assumptions - All dice are fair six-sided dice; all throws are independent. - All coins are fair; flips are independent. - The deck is a standard 52-card deck (26 red, 26 black), shuffled uniformly at random. - In the real assessment each question allows only a few seconds, so favor the fastest correct reasoning (symmetry, complements, counting) over brute-force enumeration. - Answers should be exact (e.g., $5/6$), not rough estimates. ### Clarifying Questions to Ask - Are the dice and coins fair, and are successive throws/flips independent? - Is the deck a standard 52-card deck with 26 red and 26 black cards, and is the shuffle uniformly random? - When cards are "thrown in the bin," are they discarded unseen (i.e., we learn nothing about their colors)? - Does "different face value" mean strictly different from the first throw's value? - Should answers be exact fractions, or rounded decimals to a given precision? ### Part 1 You throw one die two times. What is the probability that the second throw shows a different face value than the first throw? ```hint Condition on the first throw Whatever the first throw shows, the second throw is independent and uniform over 6 faces. How many of those 6 faces "collide" with the first result? ``` #### What This Part Should Cover - Recognizing that the answer does not depend on what the first throw actually was. - Using the complement ("same value") rather than enumerating all differing pairs. - Clean exact arithmetic with a one-line justification. ### Part 2 Take a shuffled deck of 52 playing cards and throw the top 10 cards in the bin (unseen). What is the probability that the new top card — the 11th card of the original deck — is red? ```hint Symmetry In a uniformly shuffled deck, does any fixed position (1st, 11th, 37th, ...) have a different color distribution than any other? ``` ```hint What information did you gain? The 10 discarded cards are never observed. If you learn nothing about them, can discarding them change your probability for the next card? ``` #### What This Part Should Cover - Recognizing what a uniform shuffle implies about the color distribution at any fixed position in the deck. - Reasoning carefully about whether discarding a batch of cards *unseen* can change what you know about the next card, versus the case where the discarded cards are observed. - Avoiding the trap of trying to condition on information you don't actually have. ### Part 3 You throw two dice. What is the probability that the two dice add up to 11 or 12? ```hint Count outcomes There are $36$ equally likely ordered outcomes for two dice. List the ordered pairs that sum to 11, and those that sum to 12. ``` #### What This Part Should Cover - Setting up the correct equally-likely ordered-outcome sample space for two dice. - Carefully enumerating which ordered pairs hit each target sum, without over- or under-counting (a common trap is treating an ordered pair and its reverse as the same outcome). - Reducing the count to an exact fraction. ### Part 4 You flip a coin 3 times. What is the probability that the outcome is the same for all flips (all Heads or all Tails)? ```hint Count sequences There are $2^3$ equally likely sequences. How many of them are constant? Alternatively: after the first flip, what must each subsequent flip do? ``` #### What This Part Should Cover - Setting up the full sequence sample space correctly and identifying which sequences qualify as "all the same." - Alternatively, using the equivalent conditional argument (what each flip after the first must do). - Not over- or under-counting the qualifying sequences. ### Part 5 61 coins are randomly put into 15 boxes. If any box ends up containing more than 4 coins, you win a prize. What is the probability that you win a prize? ```hint Assume the opposite Suppose no box contains more than 4 coins. What is the maximum total number of coins the 15 boxes could hold? ``` ```hint Not really a probability question This is a pigeonhole-principle question wearing a probability costume. Does the random placement mechanism matter at all? ``` #### What This Part Should Cover - Recognizing this as a counting/pigeonhole-style argument dressed up as a probability question, rather than something that requires an actual probability computation. - Comparing the total number of coins against the maximum capacity implied by "no box exceeds 4," and drawing the correct conclusion about whether the event is certain, impossible, or somewhere in between. - Noticing that the randomness of the placement mechanism is irrelevant to the conclusion here. ### What a Strong Answer Covers Across all parts, the interviewer is checking that you: - Choose the fastest valid tool for each question — complement counting, symmetry, direct enumeration over a small sample space, or a deterministic (pigeonhole) argument — rather than one heavy method for everything. - State the modeling assumptions (fairness, independence, uniform shuffle, unseen discards) before computing. - Produce exact answers with confident, error-free mental arithmetic under time pressure. - Recognize degenerate cases where the answer is $0$ or $1$ and no computation is needed. ### Follow-up Questions - In Part 2, how does the answer change if the 10 discarded cards are turned face up and you see that 7 of them are red? - In Part 5, suppose only 60 coins are placed uniformly at random into the 15 boxes. Is the probability of winning still 1? How would you reason about it? - In Part 1, what is the expected number of throws until you first see a face value that repeats an earlier one? - In Part 3, generalize: for two fair dice, which target sum $s$ maximizes $P(\text{sum} = s)$, and why?