Solution
to Surprise Appearance:
Drawing Number

Probability of
Drawing the First Black Marble

Probability of
Drawing the Second Black Marble

1^{st}

What
is the probability of drawing a black marble on the first draw?
2/10
= 9/45

It
is impossible to draw the 2^{nd} black marble on the first draw.

2^{nd}

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

3^{rd}

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 
4^{th}

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 
5^{th}

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 
6^{th}

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 
7^{th}

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 
8^{th}

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 
9^{th}

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 10^{th} (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 1^{st} 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 1^{st} Black Marble being drawn

Probability of
the 2^{nd} Black Marble being drawn

1^{st}

9/45

0/45 or 0

2^{nd}

8/45

1/45

3^{rd}

7/45

2/45

4^{th}

6/45

3/45

5^{th}

5/45

4/45

6^{th}

4/45

5/45

7^{th}

3/45

6/45

8^{th}

2/45

7/45

9^{th}

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], 347354. Via Greg Ross at www.futilitycloset.com .
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