Solution
to Surprise Appearance:
|
Drawing Number
|
Probability of
Drawing the First Black Marble
|
Probability of
Drawing the Second Black Marble
|
|
1st
|
What
is the probability of drawing a black marble on the first draw?
2/10
= 9/45
|
It
is impossible to draw the 2nd black marble on the first draw.
|
|
2nd
|
What
is the probability of drawing a white marble on the first draw, and then a
black marble on the second draw (WB)?
8/10
x 2/9 = 8/45
|
What
is the probability of drawing a black marble on the first draw, and then a
black marble on the second draw (BB)?
2/10
x 1/9 = 1/45
|
|
3rd
|
What
is the probability of drawing white white black (WWB)?
8/10
x 7/9 x 2/8 = 7/45
|
What
is the probability of drawing black white black (BWB), or white black black (WBB)?
(2/10
x 8/9 x 1/8) +
(8/10 x 2/9 x 1/8) = 2/45 |
|
4th
|
What
is the probability of drawing WWWB?
8/10
x 7/9 x 6/8 x 2/7 = 6/45
|
What
is the probability of drawing BWWB, or WBWB, or WWBB?
(2/10
x 8/9 x 7/8 x1/7) +
(8/10 x 2/9 x 7/8 x 1/7) + (8/10 x 7/9 x 2/8 x 1/7) = 3/45 |
|
5th
|
What
is the probability of drawing WWWWB?
8/10
x 7/9 x 6/8 x 5/7 x 2/6 = 5/45
|
What
is the probability of drawing BWWWB, or WBWWB, or WWBWB, or WWWBB?
(2/10
x 8/9 x 7/8 x 6/7 x 1/6) +
(8/10 x 2/9 x 7/8 x 6/7 x 1/6) + (8/10 x 7/9 x 2/8 x 6/7 x 1/6) + (8/10 x 7/9 x 6/8 x 2/7 x 1/6) = 4/45 |
|
6th
|
What
is the probability of drawing WWWWWB?
8/10
x 7/9 x 6/8 x 5/7 x 4/6 x 2/5 = 4/45
|
What
is the probability of drawing BWWWWB, or WBWWWB, or WWBWWB, or WWWBWB, or WWWWBB?
(2/10
x 8/9 x 7/8 x 6/7 x 5/6 x 1/5) +
(8/10 x 2/9 x 7/8 x 6/7 x 5/6 x 1/5) + (8/10 x 7/9 x 2/8 x 6/7 x 5/6 x 1/5) + (8/10 x 7/9 x 6/8 x 2/7 x 5/6 x 1/5) + (8/10 x 7/9 x 6/8 x 5/7 x 2/6 x 1/5) = 5/45 |
|
7th
|
What
is the probability of drawing WWWWWWB?
8/10
x 7/9 x 6/8 x 5/7 x 4/6 x 3/5 x 2/4 = 3/45
|
What
is the probability of drawing BWWWWWB, or WBWWWWB, or WWBWWWB, or WWWBWWB, or
WWWWBWB, or WWWWWBB?
(2/10
x 8/9 x 7/8 x 6/7 x 5/6 x 4/5 x 1/4) +
(8/10 x 2/9 x 7/8 x 6/7 x 5/6 x 4/5 x 1/4) + (8/10 x 7/9 x 2/8 x 6/7 x 5/6 x 4/5 x 1/4) + (8/10 x 7/9 x 6/8 x 2/7 x 5/6 x 4/5 x 1/4) + (8/10 x 7/9 x 6/8 x 5/7 x 2/6 x 4/5 x 1/4) + (8/10 x 7/9 x 6/8 x 5/7 x 4/6 x 2/5 x 1/4) = 6/45 |
|
8th
|
What
is the probability of drawing WWWWWWWB?
8/10
x 7/9 x 6/8 x 5/7 x 4/6 x 3/5 x 2/4 x 2/3 = 2/45
|
What
is the probability of drawing BWWWWWWB, or WBWWWWWB, or WWBWWWWB, or
WWWBWWWB, or WWWWBWWB, or WWWWWBWB, or WWWWWWBB?
(2/10
x 8/9 x 7/8 x 6/7 x 5/6 x 4/5 x 3/4 x 1/3) +
(8/10 x 2/9 x 7/8 x 6/7 x 5/6 x 4/5 x 3/4 x 1/3) + (8/10 x 7/9 x 2/8 x 6/7 x 5/6 x 4/5 x 3/4 x 1/3) + (8/10 x 7/9 x 6/8 x 2/7 x 5/6 x 4/5 x 3/4 x 1/3) + (8/10 x 7/9 x 6/8 x 5/7 x 2/6 x 4/5 x 3/4 x 1/3) + (8/10 x 7/9 x 6/8 x 5/7 x 4/6 x 2/5 x 3/4 x 1/3) + (8/10 x 7/9 x 6/8 x 5/7 x 4/6 x 3/5 x 2/4 x 1/3) = 7/45 |
|
9th
|
What
is the probability o drawing WWWWWWWWB?
8/10
x 7/9 x 6/8 x 5/7 x 4/6 x 3/5 x 2/4 x 1/3 x 2/2 = 1/45
|
What
is the probability of drawing BWWWWWWWB, or WBWWWWWWB, or WWBWWWWWB, or WWWBWWWWB,
or WWWWBWWWB, or WWWWWBWWB, or WWWWWWBWB, or WWWWWWWBB?
(2/10
x 8/9 x 7/8 x 6/7 x 5/6 x 4/5 x 3/4 x 2/3 x 1/2) +
(8/10 x 2/9 x 7/8 x 6/7 x 5/6 x 4/5 x 3/4 x 2/3 x 1/2) + (8/10 x 7/9 x 2/8 x 6/7 x 5/6 x 4/5 x 3/4 x 2/3 x 1/2) + (8/10 x 7/9 x 6/8 x 2/7 x 5/6 x 4/5 x 3/4 x 2/3 x 1/2) + (8/10 x 7/9 x 6/8 x 5/7 x 2/6 x 4/5 x 3/4 x 2/3 x 1/2) + (8/10 x 7/9 x 6/8 x 5/7 x 4/6 x 2/5 x 3/4 x 2/3 x 1/2) + (8/10 x 7/9 x 6/8 x 5/7 x 4/6 x 3/5 x 2/4 x 2/3 x 1/2) + (8/10 x 7/9 x 6/8 x 5/7 x 4/6 x 3/5 x 2/4 x 2/3 x 1/2) = 8/45 |
|
10th
|
It is impossible
to draw the first black marble on the 10th (or last) drawing.
|
What
is the probability of drawing BWWWWWWWWB, or WBWWWWWWWB, or WWBWWWWWWB, or
WWWBWWWWWB, or WWWWBWWWWB, or WWWWWBWWWB, or WWWWWWBWWB, or WWWWWWWBWB, or
WWWWWWWWBB?
(2/10
x 8/9 x 7/8 x 6/7 x 5/6 x 4/5 x 3/4 x 2/3 x 1/2 x 1/1) +
(8/10 x 2/9 x 7/8 x 6/7 x 5/6 x 4/5 x 3/4 x 2/3 x 1/2 x 1/1) + (8/10 x 7/9 x 2/8 x 6/7 x 5/6 x 4/5 x 3/4 x 2/3 x 1/2 x 1/1) + (8/10 x 7/9 x 6/8 x 2/7 x 5/6 x 4/5 x 3/4 x 2/3 x 1/2 x 1/1) + (8/10 x 7/9 x 6/8 x 5/7 x 2/6 x 4/5 x 3/4 x 2/3 x 1/2 x 1/1) + (8/10 x 7/9 x 6/8 x 5/7 x 4/6 x 2/5 x 3/4 x 2/3 x 1/2 x 1/1) + (8/10 x 7/9 x 6/8 x 5/7 x 4/6 x 3/5 x 2/4 x 2/3 x 1/2 x 1/1) + (8/10 x 7/9 x 6/8 x 5/7 x 4/6 x 3/5 x 2/4 x 2/3 x 1/2 x 1/1) + (8/10 x 7/9 x 6/8 x 5/7 x 4/6 x 3/5 x 2/4 x 1/3 x 2/2 x 1/1) = 9/45 |
It is interesting to see that the first
black marble is more likely to be drawn on the first drawing than any other
drawing. And that the probability of
drawing the 1st black marble on the second or subsequent drawing
consistently decreases. The probability
of drawing the second black marble follows the same pattern in reverse order.
|
Drawing Number
|
Probability of
the 1st Black Marble being drawn
|
Probability of
the 2nd Black Marble being drawn
|
|
1st
|
9/45
|
0/45 or 0
|
|
2nd
|
8/45
|
1/45
|
|
3rd
|
7/45
|
2/45
|
|
4th
|
6/45
|
3/45
|
|
5th
|
5/45
|
4/45
|
|
6th
|
4/45
|
5/45
|
|
7th
|
3/45
|
6/45
|
|
8th
|
2/45
|
7/45
|
|
9th
|
1/45
|
8/45
|
|
10th
|
0/45 or 0
|
9/45
|
|
Total Probability
|
45/45 or 1
|
45/45 or 1
|
Source: A. E. Lawrence, “Playing With
Probability,” Mathematical Gazette, vol. 53 [December 1969], 347-354. Via Greg Ross at www.futilitycloset.com .
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