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中文题目Six distinguishable balls are thrown independently and uniformly at random into 4 distinguishable urns. What is the probability that at least one urn contains 3 or more balls? Give an exact fraction.
打开 →If a logistic model outputs score z = ln 3, what probability does it assign to class 1?
打开 →A 5-card poker hand is dealt from a standard 52-card deck. What is the probability that all five cards are of the same suit (a flush, including straight flushes)?
打开 →A 5-card poker hand is dealt from a standard 52-card deck. What is the probability of being dealt a full house (three cards of one rank and two cards of another rank)?
打开 →A 5-card hand is dealt from a standard 52-card deck. What is the probability that the hand is a straight but not a straight flush (i.e., five cards of consecutive ranks, not all of the same suit)? Aces can be high or low (A-2-3-4-5 through 10-J-Q-K-A).
打开 →From state A, the jump rates to B and C are 0.4 and 0.6. What is the probability that by time 1 either no jump has happened yet or, if a jump has happened, the first jump was to C?
打开 →A 5-card poker hand is dealt from a standard 52-card deck. What is the probability that the hand contains exactly one pair (two cards of one rank and three cards of three different other ranks, with no other matching ranks)?
打开 →From state A, the jump rates to B and C are 1.5 and 0.5. What is the probability that no jump has occurred by time 0.4?
打开 →Suppose 25 independent null strategies each produce an in-sample Sharpe that is approximately standard normal. What is the probability that the best observed Sharpe exceeds 1.5?
打开 →From state A, the jump rates to B and C are 0.8 and 1.2. What is the probability that by time 0.5 a jump has occurred and that first jump was to B?
打开 →From state A, the jump rates to B, C, and D are 0.6, 0.9, and 0.5. What is the probability that by time 1 the first jump has occurred and it was to D?
打开 →From state A, the jump rates to B, C, and D are 0.3, 0.5, and 0.2. What is the probability that by time 2 the first jump has occurred and it was to C?
打开 →A Bernoulli success probability has prior $\mathrm{Beta}(3,5)$. After observing 6 successes and 2 failures, compute the requested posterior predictive probability for the next 1 trial(s).
打开 →Leverage L takes values 1 and 4. If E[L] = 2.2, what probability p is on L=1, and what is E[1/(1+L)]?
打开 →Long inventory, you skew quotes to sell. The ask fills with probability 0.45 earning net edge 0.020 per share; the bid fills with probability 0.10 but is toxic, earning net edge -0.030 per share. Each side quotes 100 shares. Assuming at most one side fills, what is the expected P
打开 →Let $F_X$ be a continuous, strictly increasing CDF and $U \sim \operatorname{Uniform}(0,1)$. Prove that $Y = F_X^{-1}(U)$ has CDF $F_X$. Conversely, show that if $X$ has CDF $F_X$, then $F_X(X) \sim \operatorname{Uniform}(0,1)$.
打开 →Let $X_1, X_2, X_3, X_4$ be independent $\operatorname{Uniform}(0,1)$ random variables. Compute $P(X_{(3)} < 0.5)$, where $X_{(3)}$ is the third smallest.
打开 →A portfolio consists of $n = 50$ stocks with equal weight $1/n$. The annual returns $R_1, \ldots, R_{50}$ are independent, each with mean $\mu = 0.08$ (i.e., $8\%$) and standard deviation $\sigma = 0.20$. The portfolio return is $\bar{R} = \frac{1}{50}\sum_{i=1}^{50} R_i$. **(a
打开 →Let $X_1, X_2, X_3$ be independent $\operatorname{Uniform}(0,1)$ random variables. The range is $R = X_{(3)} - X_{(1)}$. Compute $P(R > \tfrac{1}{2})$.
打开 →The starting point X is uniform on [0,8]. Brownian motion starts at X and stops on first exit from [0,8]. What is the average probability of exiting through 8?
打开 →Five distinguishable balls are thrown independently and uniformly at random into 12 distinguishable urns. What is the probability that at least two balls land in the same urn? Give an exact fraction.
打开 →A one-step trinomial tree has multipliers u=1.2, m=1, d=0.8. The middle probability is fixed at p_m=0.6, r=0.04, Δt=1. Find the up-move probability p_u that makes the discounted underlying a martingale (so p_u+p_m+p_d=1 and E[S_1]=S_0 e^{rΔt}).
打开 →Each individual has $0$ children with probability $0.3$ and $2$ children with probability $0.7$. Starting from one ancestor, compute the extinction probability.
打开 →If the extinction probability from one ancestor is $q$, what is the extinction probability from $k$ independent ancestors?
打开 →A stock has spot 100, strike 100, rate 0.03, dividend yield 0.01, volatility 0.2, and maturity 1. Under Black-Scholes, what are the forward price F_0,T and the risk-neutral probability that the call finishes in the money?
打开 →A stock has spot 95, strike 100, rate 0.04, dividend yield 0.02, volatility 0.25, and maturity 0.5. Under Black-Scholes, what are the forward price F_0,T and the risk-neutral probability that the call finishes in the money?
打开 →A stock has spot 120, strike 110, rate 0.02, dividend yield 0, volatility 0.18, and maturity 1.5. Under Black-Scholes, what are the forward price F_0,T and the risk-neutral probability that the call finishes in the money?
打开 →For an even-money bet you are unsure of the true win probability: it is equally likely to be 0.52 or 0.62. A colleague plugs the average estimate 0.57 into the Kelly formula and bets 0.14 of capital. To maximize expected log growth you should instead average the realized growth o
打开 →Brownian motion starts at x = 1 and has upper barrier 4 and lower barrier a < 1. What a makes the probability of hitting 4 before a equal to 0.4?
打开 →An event is quoted at 3-to-1 odds against. What is the implied probability that it happens?
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