PASCAL'S TRIANGLE

$triangle$

1
1    1
1    2    1
1    3    3    1
1    4    6    4    1
1    5    10 10    5    1

& SO ON....

~To get a number, just add the two numbers directly above, except for the 1's on the ends. Reading a row, gives you the combinations of the 2nd number (left to right), taken 0, 1, 2, 3, ... at a time.

~For example,
in the above triangle, the 3rd number in the bottom row formed, is 10 (from the left). This will equal 5C2 . It turns out the 5C3 also = 10.

~Also, we can use Pascal's triangle to calculate binomial probabilities.

~To do this, note that the sum of the numbers in any given row equals 2ⁿ, where n is always the 2nd number from the left (or right). (i.e., for the numbers 1, 5, 10, 10, 5, 1, we have, 1+5+10+10+5+1=32=2⁵) (here n = 5)

~So, if we are interested in the probability of getting at least 6 heads when tossing 7 coins, we would have to expand our triangle two more rows so the 2nd number on the right is n=7. That would give us the following row:
1 7 21 35 35 21 7 1.

~Now for P(at least 6 heads in 7 tosses) = (7 + 1)/ 2⁷ = 8/128 or .0625

~Notice that we took the sum of the last 2 numbers in that row since they give us the nimber of ways we can get 6 or 7 heads, respectively. Also note that 2⁷ represents the total possible number of outcomes in the sample space when tossing 7 coins.

~Obviously, if the n (number of trials) is quite larger (say n > 20), using Pascal's triangle is laborious, since your triangle would be very wide & it would be a tedious job to construct.

~Therefore, the use of the TI-83 is recommended

~However, just in case you are isolated on an island without a calculator, this could be a nice way to occupy your time.