How difficult is to find a ford?

During the Bunzlau scenario, the Karpov's Cossacks were trying to find a ford upstream the watermill. If they had found it, they could have crossed the river Bober outflanking the French defense.
Although the capabilities of irregular cavalry units in Lasalle are quite limited (despite my home rule), the presence of this force in the area had surely hampered the French defense of the bridge and would have contributed, in combination with the frontal attack of Prussian infantry, to the Allied victory.
To find a ford, the Cossacks were to be in contact with the river for a full turn without doing anything else, and roll a die. A result equal or greater than 4 would mean that a ford had been found, and the Cossacks could have passed the river the next turn. However, they were seeking for a ford for 6 consecutive turns with negative results!
The chances of such an event can be found by means of statistics. The problem is based on the binomial distribution, B(n,p), that gives the probability of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. In our case, n=6 and p=0,5 (a ford is found by rolling 4, 5 or 6 with 1D6). Using the adequate formula, the probability to achieve 6 failures in 6 successive rolls can be calculated as 0,5^6 = 0,015625 = 1,56%.
The Cossacks were really unlucky!

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