F.7 Probability [20 points]

2011-09-11 5:58 am

1. Four students leave umbrellas on the umbrella-stand of the university library. Absent-mindedly, they pick umbrellas at random when they leave. Show that the chance that no one gets his own umbrella is 3/8.
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2. A pair of fair dice is tossed. Find the probability that the maximum of the two numbers is greater than 4.
(Ans. 5/9)
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3. A table of random numbers is built up from the digits 0 to 9 =, each digit being chosen at random. Find the probability that

a) the digit 4 appears exactly once in a two-digit number
b) the digit 4 appears exactly once in a three-digit number
c) only two zeros appear in a three-digit number

( Ans for
a = 0.18
b = 0.243
c = 0.027)

回答 (1)

用戶2053

2011-09-11 6:36 am

✔ 最佳答案

1)1 2 3 4
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For the first one is 2 , 3 cases as follow :
2 1 4 3
2 3 4 1
2 4 1 3Similarly , there are 3 cases for the first one is 3 or 4 , Total 3 x 3 = 9 casesP(E) = 9 / 4! = 3/8
2)P(E)= 1 - P(The number of the first dice < 5) xP(The number of the second dice < 5) = 1 - (4/6)(4/6) = 5/9
3a)P(The digit 4 appears exactly once in a two-digit number)
= P(4?) + P(?4) ....... (? is not 4)
= 0.1 * 0.9 + 0.9 * 0.1
= 0.18b)P(The digit 4 appears exactly once in a three-digit number)
= P(4??) + P(?4?) + P(??4) ....... (? is not 4)
= 3 x 0.1 * 0.9 * 0.9
= 0.243c)P(Only two zeros appear in a three-digit number)
= P(?00) + P(0?0) + P(00?)....... (? is not 0)
= 3 (0.9 * 0.1 * 0.1)
= 0.027

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