I was wondering if someone could please explain these probability formulae:
in language a 2U student could understand
-Permutations
nP
n=n!
-binomial probability distribution P(r successes) = <sup>n</sup>C
rp<sup>r</sup>q<sup>n-r</sup>
<sup></sup>
Examples would also be helpful
Thanks
I'll try to explain:
1) say you have 5 distinct objects A,B,C,D and E. To find how many ways we can arrange these 5 objects amongst themselves let's imagine we have 5 boxes into which we can assign them, one per box. The 1st box can be filled in 5 ways; it can be A, it can be B etc. Having dealt with the 1st box the 2nd box can be filled in 4 ways. In this way the remaining 3 boxes can be filled in 3, in 2 and in 1 ways respectively. All in all the 5 boxes can be filled in 5x4x3x2x1 (i.e. 5!) ways. By a similar reasoning n distinct objects can be arranged amongst themselves in n! ways; i.e sub[n][/sub]P
n = n!
2) suppose there is a 1-in-5 chance of drawing a red ball from a jar; then the chance of not drawing a red ball is 1 - 1/5 = 4/5 = 0.8. If drawing a red ball is what we are after, then drawing a red from the jar would be referred to as a 'success'; the converse outcome, i.e. drawing a ball that is not red would be called a failure. So for each draw of a ball from the jar,
p = prob(success) = 0.2 and
q = prob(failure) = 0.8. Imagine we repeat this draw a total of 10 times, returning the ball to the jar each time (and presumably stirring the contents). In binomial probability, we call each draw an 'experiment'. In this example we are repeating the experiment 10 times. We assume the outcome of each experiment (a red or not a red) is not affected by the results of the previous draws. At the end of the 10 draws we can get 0 red, 1 red, 2 reds, . . . 10 reds. The outcome '3 reds' can occur in
10C
3 ways: each with prob
pxpxpxqxqxqxqxqcxqxq =
p3q7 as follows: rrrnnnnnnn, rrnrnnnnnn . . . . nnnnnnnrrr (where n stands for 'not red'). In this way you can find the prob of 0 red, of 1 red, of 2 reds . . . of10 reds. Now the 11 probabilities are precisely the terms of the binomial expansion: (
p +
q)
10
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