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You can put this solution on YOUR website!
\n" ); document.write( "This is a great question. Consider we have a 4x4 grid as shown below
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| X | A1 | A2 | A3 |
| A4 | A5 | A6 | A7 |
| A8 | A9 | A10 | A11 |
| A12 | A13 | A14 | A15 |
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\n" ); document.write( "\n" ); document.write( "There are 16 items total. One of which is X. The other 15 are A1 through A15. The probability of randomly selecting X is 1/16\r
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\n" ); document.write( "\n" ); document.write( "Now consider this grid
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| Y | B1 | B2 |
| B3 | B4 | B5 |
| B6 | B7 | B8 |
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\n" ); document.write( "\n" ); document.write( "We have 9 items. One is Y, the other 8 are B1 through B8. The probability of randomly selecting Y is 1/9\r
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\n" ); document.write( "\n" ); document.write( "I claim that the probability of selecting both X and Y at the same time, again completely randomly, is 1/144 which is the result of multiplying 1/16 and 1/9. Both events are independent as neither grid interacts with one another. \r
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\n" ); document.write( "\n" ); document.write( "This is not obvious why this is the case. But let's say we arranged the items in the first grid to be along the left side of a very large table (16 rows) and the items of the second grid to be along the top of the large table (9 columns). We have this blank table
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| Y | B1 | B2 | B3 | B4 | B5 | B6 | B7 | B8 | |
| X | |||||||||
| A1 | |||||||||
| A2 | |||||||||
| A3 | |||||||||
| A4 | |||||||||
| A5 | |||||||||
| A6 | |||||||||
| A7 | |||||||||
| A8 | |||||||||
| A9 | |||||||||
| A10 | |||||||||
| A11 | |||||||||
| A12 | |||||||||
| A13 | |||||||||
| A14 | |||||||||
| A15 |
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\n" ); document.write( "\n" ); document.write( "I'll fill out part of the table. The first row shows
\n" ); document.write( "XY, XB1, XB2, ... , XB8
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| Y | B1 | B2 | B3 | B4 | B5 | B6 | B7 | B8 | |
| X | XY | XB1 | XB2 | XB3 | XB4 | XB5 | XB6 | XB7 | XB8 |
| A1 | |||||||||
| A2 | |||||||||
| A3 | |||||||||
| A4 | |||||||||
| A5 | |||||||||
| A6 | |||||||||
| A7 | |||||||||
| A8 | |||||||||
| A9 | |||||||||
| A10 | |||||||||
| A11 | |||||||||
| A12 | |||||||||
| A13 | |||||||||
| A14 | |||||||||
| A15 |
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\n" ); document.write( "\n" ); document.write( "Notation:
\n" ); document.write( "XY = we selected X and Y
\n" ); document.write( "XB1 = we selected X and B1
\n" ); document.write( "XB2 = we selected X and B2
\n" ); document.write( "etc etc\r
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\n" ); document.write( "\n" ); document.write( "The rest of the table is filled out the same way. You just use the headers as shown. Anyways, note how there are 16 rows and 9 columns giving 16*9 = 144 items total in this massive table. Ask yourself: how many items are XY? That would be one only. This shows the probability of getting XY randomly is 1/144.\r
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\n" ); document.write( "\n" ); document.write( "As you can see, this can be thought of as an area problem. Areas of rectangles are found by multiplying the length and width. Squares are a special type of rectangle. Consider a square of area 1. This represents 100% probability.\r
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\n" ); document.write( "\n" ); document.write( "Draw a square with area 1. Mark the upper half as a blue rectangle. If we throw a dart, then the probability of landing in the blue shaded region is 1/2. This is because 1/2 of the area is blue, and the total area is 1, so (1/2)/1 = 1/2.
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![](\"https://i.imgur.com/r3KifXu.png\")
\n" ); document.write( "Square divided into 2 equal pieces\r
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\n" ); document.write( "\n" ); document.write( "Draw another square of area 1. Divide the rectangle into 3 vertical equal slices (think of a candy bar being broken up perhaps).
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![](\"https://i.imgur.com/kykd4p4.png\")
\n" ); document.write( "Square divided into 3 equal pieces. The probability of landing in the red region is 1/3.\r
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\n" ); document.write( "\n" ); document.write( "If we overlap the two figures, then we'll have the blue and red regions combine to form the purple area
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![](\"https://i.imgur.com/giWhNnC.png\")
\n" ); document.write( "Square divided into 6 equal pieces\r
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\n" ); document.write( "\n" ); document.write( "Ask yourself this: what is the probability of landing in BOTH the upper blue area and the red area at the same time? That would be 1/6. The area of the blue rectangle is (1/2)*1 = 1/2. The area of the red rectangle is 1*(1/3) = 1/3. So the purple region has area (1/2)*(1/3) = 1/6\r
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\n" ); document.write( "\n" ); document.write( "Or you can think of it like this: Start with a red rectangle (area 1/3) and cut it in half to get the purple rectangle (area 1/6). A slight variation: Start with a blue rectangle (area 1/2) and cut that into thirds to get the purple piece (area 1/6)
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