Mathematics - Probability

(a)
P(1st ball is multiple of 4)
= P(1st ball is 4/8/12 /16/20)
= 5/20
= 1/4

P(1st ball is multiple of 4 and 2nd ball is multiple of 5)
= P(1st ball is 4/8/12/16, 2nd ball is 5/10/15/20) + P(1st ball is 20, 2nd ball is 5/10/15)
= (4/20) x (4/19) + (1/20) x (3/19)
= (16/380) + (3/380)
= 19/380
= 1/20

P(2nd ball is multiple of 5 | 1st ball is multiple of 4)
= P(1st ball is multiple of 4 and 2nd ball is multiple of 5) / P(1st ball is multiple of 4)
= (1/20) / (1/4)
= 1/5



(b)
P(1st ball is multiple of 4 and 2nd ball is multiple of 8)
= P(1st is 4/12/20, 2nd is 8/16) + P(1st is 8, 2nd is 16) + P(1st is16, 2nd is 8)
= (3/20) x (2/19) + (1/20) x (1/19) + (1/20) x (1/19)
= (6/380) + (1/380) + (1/380)
= 8/380
= 2/95

P(2nd ball is multiple of 8 | 1st ball is multiple of 4)
= P(1st ball is multiple of 4 and 2nd ball is multiple of 8) / P(1st ball is multiple of 4)
= (2/95) / (1/4)
= 8/95
=

(a) A ball of multiple of 4 is drawn. The number of balls that satisfy the second condition is affected if and only if the ball drawn is 20 since 20 is the only common multiple of 4 and 5 in the 1 - 20 range.

There are only 19 balls after the first draw. There is a 1/5 probability of drawing 20 (out of 4, 8, 12, 16, 20), and 4/5 of not drawing a 20. The number of balls satisfying the condition becomes 3 when the first ball is 20 (5, 10, 15 are left), while it's no change (Still 4) when the first ball is not 20.

P (Second ball is a multiple of 5 | First ball is a multiple of 4)
= P (Second ball is a multiple of 5 | First ball is 20) + P (Second ball is a multiple of 5 | First ball is a multiple of 4 excluding 20)

= (1/5) (3/19) + (4/5) (4/19) = 1/5

(b) Similar to (a)

P (Second ball is a multiple of 8 | First ball is a multiple of 4)
= P (Second ball is a multiple of 8 | First ball is 8 or 16) + P (Second ball is a multiple of 8 | First ball is a multiple of 4 excluding 8 and 16)

= (2/5) (1/19) + (3/5) (2/19) = 8/95

the first ball drawn is a multiple of 4 (4,8,12,16,20)
(a) a multiple of 5, (5,10,15,20)
(b) a multiple of 8. (8,16)

a) (4/20)(4/19) + (1/20)(3/19)= 1/20
(4/20) 係第1次抽個時, 抽唔中 20,
(4/19) 係第2次抽中 5的倍數(5,10,15,20)的機率
(1/20) 係第1次抽個時, 抽中20
(3/19) 係第2次抽個時, 係5的倍數, 但無左20



b) (3/20)(2/19)+(2/20)(1/19)= 2/95
(3/20) 係第1次抽到 4, 12, 20的機率
(2/19) 係第2次抽到 8 或 16
(2/20) 係第1次抽到 8 或 16機率
(1/19) 係第2次抽到 8 或 16機率(因為第1次抽左其中一個)

2009-01-19 00:55:00 補充：
a) 1/20 = 0.05

b) 2/95 = 0.0211

a) The first ball is mutiple of 4, that mean 1/4 is 20;
So probability of the second ball is a mutiple of 5 is: 1/4x3/19+3/4x4/19= 15/76;
b) The first ball is mutiple of 4, that mean 2/5 is 8, 16;
So probability of the second ball is a mutiple of 8 is: 2/5x1/19+3/5x2/19=8/95;

These are the answer.

2009-01-22 22:27:18 補充：
Yes, you are right, so
a) The first ball is mutiple of 4, that mean 1/5 is 20;
So probability of the second ball is a mutiple of 5 is: 1/5x3/19+4/5x4/19= 19/95=1/5;