题目

有两道数学关于概率论的题目不太会 题目是很简单的英文 我就不翻译了,求高人帮助!
1. I form a 3-digit number at random from the digits 1,2,3,..., 9; determine (work your answers out completely) the probability that my number has ...
(a) No repeated digits.
(b) All three digits the same.
(c) Exactly one repeated digit (i.e. two digits the same and one dierent).
Don't forget to work your answers out completely.
2. A bag contains 10 marbles: 1 red, 2 blue, 3 green, and 4 yellow. I reach into the bag and pick 5 marbles at random. Determine the probability that my pick has ...
(a) At least 2 yellow marbles.
(b) At least 2 yellow marbles and at least 1 green marble.
(c) All four colors represented.
(d) At least two colors represented.
(e) At most three colors represented.
Leave your answers in factorial notation.

提问时间：2021-01-27

答案

1.
(a) No repeated digits. P(9,3)/9^3=9*8*7/9^3=56/81
(b) All three digits the same. C(9,1)/9^3=9/9^3=1/81
(c) Exactly one repeated digit. 1- 56/81-1/81=24/81=8/27

2.
(a) At least 2 yellow marbles.
1-C(6,5)/C(10,5)-C(4,1)*C(6,4)/C(10,5)=1-1/42-10/42=31/42
(b) At least 2 yellow marbles and at least 1 green marble.
1-C(7,5)/C(10,5)-C(6,5)/C(10,5)-C(4,1)*C(6,4)/C(10,5)
=1-7/84-1/42-10/42
=55/84
(c) All four colors represented.
C(1,1)*C(2,1)*C(3,1)*C(4,1)/C(10,5)
=1*2*3*4*(1*2*3*4*5)/(10*9*8*7*6)
=2/21
(d) At least two colors represented.
1-C(9,5)/C(10,5)-C(8,5)/C(10,5)-C(7,5)/C(10,5)-C(6,5)/C(10,5)
=1-1/2-2/9-1/12-1/42
=209/252
(e) At most three colors represented.
1-C(1,1)*C(2,1)*C(3,1)*C(4,1)/C(10,5)
=1-2/21
=19/21