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 ```premium-lock What This Part Should Cover ``` ### 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 ```premium-lock What This Part Should Cover ``` ### 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 ```premium-lock What This Part Should Cover ``` ### 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 ```premium-lock What This Part Should Cover ``` ### 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 ```premium-lock What This Part Should Cover ``` ### What a Strong Answer Covers ```premium-lock What a Strong Answer Covers ``` ### 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?