Processing math: 61%

考試

107成大資聯離散[參考解答]

1. Suppose we randomly choose nonnegative integers x1,x2,x3,and x4 that solve the equation x1+x2+x3+x4=10. We assume that each solution has an equal probability of being chosen. Given that at least one of x1 and x2 is equal to 2, what is the probability that x2=2? (15%)

參考解答:4583


2. Suppose that Mark selects a ball by first picking one of two boxes at random and then selecting a ball from this box at random. The first box contain 5 red balls and 4 blue balls, and the second box contain 3 red balls and 6 blue balls. What is the probability that Mark picked a ball from the second box if he has selected a red ball? (15%)

參考解答:38


3. Solve the following recurrence relation:

3a_n – 6a_{n-1} – 3a_{n-2} + 6a_{n-3} = 0

with a_0 = 1, \, a_1 = 0, \, a_2 = 7. (10%)

參考解答:a_n = \dfrac{-5}{2} + \dfrac{3}{2}(-1)^n + 2^{n+1}, \, n \geq 0.


4. Find the set of all solutions x to the system of congruences: (10%)

x \equiv 4(\text{mod }5) \quad \text{and} \quad x \equiv 5(\text{mod }15)

參考解答:The system has no solution.


試題(pdf):連結

有任何問題,或想要完整解釋的,歡迎在下方留言唷!

發佈留言