ISCAM
Lab 2

1: Background 2: Files Needed 3: Summarizing data 4: Drawing conclusions beyond the data 5: Proability model 6: Probability model cont 7: Probability model cont 8: Conclusions

Probabilty model cont.

Binomial probabilities

When we are willing to assume we have a binomial random process, we can apply a few probability rules to help us calculate different probabilities. Under the null model, we assume that the number of "top down" outcomes in the toast dropping process will follow a binomial distribution with n = 10 and probability of success = 0.50.

For example, to find the probability of the outcome: SSSFSSFSSFF,

Under the null model, P(S) = 0.50	P(success) = 0.50
The trials are independent so we can multiply these probabilities together.	$P(SSSFSSFSSFF)\; =\; (.5)...\; (.5)\; =\; (.5)10$
We can count all of the possible outcomes have 6 successes and 4 failures = "10 choose 4"	$C(10,4)\; =\; 10!/(6!\; 4!)\; =\; 210$
These 210 outcomes are mutually exclusive so we can add together the probabilities for all of those possible outcomes with 4 successes.	210 x (.510) = .205

In other words, P(4 successes and 6 failures in 10 tosses) = 0.205.

But this isn't quite our p-value yet. To be the p-vaue we want P(4 or fewer successes in 10 tosses).

$P(4\; or\; fewer)\; =\; C(10,4)(.510)\; +\; C(10,3)(.510)\; +\; C(10,2)(.510)\; +\; C(10,1)(.510)\; +\; C(10,0)(.510)$

Which a calculator or computer will tell us equals 0.3770.

In general

Let π represent the probability of success and x the number of successes:

$P(X\; =\; x)\; =\; C(n,\; x)x\; \pi x(1-\pi )n-x$