# CAS and Probability

A recent thread on the TI-Nspire Google Group asked about uses of CAS in probability.  There are so many possibilities–one uses CAS for binomial probabilities. For example, what’s the probability of getting exactly 3 heads in 5 tosses of a fair coin?  A CAS approach expands $(\frac{1}{2}h+\frac{1}{2}t)^5$.  The $\frac{1}{2}$ coefficients of h (heads) and t (tails) are the respective probabilities of each outcome and the exponent is the number of trials. Obviously, there’s lots to unpack here to prevent this from being a black box tool, but note the power of the output.  The three heads event is represented by the $\frac{5h^3t^2}{16}$ term, and the coefficient is the desired probability, $\frac{5}{16}$. Early in my career, I taught this by expanding $(h+t)^5$, picking the appropriate term, and substituting for each variable its probability.  The great power of this approach is that the meaning of each fractional term remains by the presence of the variables while you gain the answers simultaneously.  Also note that while the problem asked only for the probability of exactly 3 heads, the CAS output gives the result of every possibility in the entire sample space. Variations 1)  What is the probability of 3 heads in five tosses if the coin was bent in a way that $P(h)=0.4$?  Adjust the coefficients to get 0.2304. 2)  The technique is not restricted binomial probabilities.  If there are three possible outcomes (a, b, and c) where $P(a)=0.4$, $P(b)=0.35$, and $P(c)=0.25$, then what is the probability of exactly 2 as in 3 trials? Because only 2 outcomes are specified for the 3 trials, the third could be either b or c.  These two outcomes are highlighted above, giving a total probability of 0.288. While these values certainly could be computed without a CAS, the point here is to use technology for computations, freeing users to think.