## Pascal’s Triangle

When you have a large number of trials, it is hard to work out the number of possible ways that the ‘x’ desired successes in ‘n’ trials can occur.  You can use something called Pascal’s triangle to aid you.

Pascal’s triangle is a triangle with ‘1’s all along the outsides.  The first row is a ‘1’.  The second row is two ‘1’s.  The third row has ‘1’s on the outside, and its middle number is the sum of the two numbers above it on either side – .  In general, except for the edge numbers, which are ‘1’s, any number is the sum of the two numbers above it on either side.  Here are the first few lines:

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You use the second row for problems with 1 trial, the third for problems with 2 trials, the fourth for problems with 3 trials, and so on.  The leftmost number in any row represents the number of ways you can achieve 0 successes.  The next number across represents the number of ways you can achieve 1 success.  The next number across represents the number of ways you can achieve 2 successes, and so on.

 I toss a coin 5 times.  What is the probability of me obtaining exactly 4 heads? Solution Firstly, you work out the success and failure probabilities – they are both 0.5.  Next, we want 4 successes exactly, so x = 4. To work out the number of ways 4 successes can occur in 5 trials, we look at the sixth row of Pascal’s triangle ­ – 1, 5, 10, 10, 5, 1. We are interested in 4 successes, so we look at the fifth number across – 5.  So there are 5 ways this can occur.