## Thursday, August 6, 2015

### Solution to Suprise Appearance

8/06/2015

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 .

David