For this probability question, should I consider him stepping back and then forward again? "A delirious man stands on the edge of a cliff and takes random steps either towards or away from the cliff’s edge. The probability of him stepping away from the edge is $\frac{3}{5}$ , and towards the edge is $\frac{2}{5}$ Find the probability he does not step over the cliff in his first four steps." I considered it to just be a binomial probability, so I just brought the probability of him not stepping over the cliff to the power of 4. However that gives $\frac{81}{625}$ and the answer is supposed to be $\frac{63}{125}$

Split it into **disjoint** events, and add up their probabilities:

* $P(BBBB)=0.6\cdot0.6\cdot0.6\cdot0.6$
* $P(BBBF)=0.6\cdot0.6\cdot0.6\cdot0.4$
* $P(BBFB)=0.6\cdot0.6\cdot0.4\cdot0.6$
* $P(BBFF)=0.6\cdot0.6\cdot0.4\cdot0.4$
* $P(BFBB)=0.6\cdot0.4\cdot0.6\cdot0.6$
* $P(BFBF)=0.6\cdot0.4\cdot0.6\cdot0.4$



* * *

UPDATE:

As soon as we have $2$ Bs, the man will not fall off the cliff in the first $4$ steps.

So we can simplify the solution above by splitting it into the following events:

* $P(BB )=0.6\cdot0.6$
* $P(BFB)=0.6\cdot0.4\cdot0.6$