HMMT 十一月 2013 · 团队赛 · 第 1 题
HMMT November 2013 — Team Round — Problem 1
题目详情
英文原题
- [ 3 ] Tim the Beaver can make three different types of geometrical figures: squares, regular hexagons,
and regular octagons. Tim makes a random sequence F , F , F , F , . . . of figures as follows:
0 1 2 3
• F is a square.
• For every positive integer i , F is randomly chosen to be one of the 2 figures distinct from F0
i i − 1
(each chosen with equal probability ).1
• Tim takes 4 seconds to make squares, 6 to make hexagons, and 8 to make octagons. He makes 2
one figure after another, with no breaks in between.
Suppose that exactly 17 seconds after he starts making F , Tim is making a figure with n sides. Whatis the expected value of n ?0
解析
英文解析
- [ 3 ] Tim the Beaver can make three different types of geometrical figures: squares, regular hexagons,
and regular octagons. Tim makes a random sequence F , F , F , F , . . . of figures as follows:
0 1 2 3
• F is a square.
• For every positive integer i , F is randomly chosen to be one of the 2 figures distinct from F0
i i − 1
(each chosen with equal probability ).1
• Tim takes 4 seconds to make squares, 6 to make hexagons, and 8 to make octagons. He makes 2
one figure after another, with no breaks in between.
Suppose that exactly 17 seconds after he starts making F , Tim is making a figure with n sides. Whatis the expected value of n ?0
Answer: 7 We write F = n as shorthand for “the i th figure is an n -sided polygon.”
If F = 8, the F = 6 or F = 4. If F = 6, Tim is making a 6-gon at time 13 (probability contributioni
1 2 2 2
1 / 4). If F = 4, F = 6 or F = 8 will take the time 13 mark (1 / 8 contribution each).
2 3 3
If F = 6, F = 8 or F = 4. If F = 8, it takes the 13 mark (1 / 4 contribution). If F = 4, F = 6 or
1 2 2 2 2 3
F = 8 will take the 13 mark (1 / 8 contribution each).
1 1 1 1 1 13
Thus, the expected value of the number of sides at time 13 is 0(4) + ( + + )(6) + ( + + )(8) = 7.
4 8 8 8 4 8