Input in the top-box is the probability of expecting zero events, the value in the Expected box can be any (decimal) number larger than one representing the average number of events expected. The expected can also be entered as the expected probability of an event to occur, a value between zero and one. Input for the number Observed and for the Sample size box must be integer values, positive number without decimals. Check classical to get the well known binomial distribution. If you check always mean the expected is presumed always to be a mean.
An example: In a hospital you count the number of locations with a laceration on victims from traffic accidents. You expect very few patients to have zero lacerations but that having one or more locations with a laceration follows a chance distribution. According to protocol you check fifteen locations on patients who are in the A&E department. The input is the small proportion of patients with zero locations with a laceration in the top box, the mean number of locations with a laceration in the expected box, and the number fifteen for the sample size, i.e. the number of locations seen on each patient. The expected input can also be the probability for patients of having one or more locations with a laceration, a number between zero and one. Dependent on what you put into the observed box the program echoes the probability of having, one, two, three or more locations with a laceration and some further statistics.
Philosophically this distribution has exactly the same properties as the usual Binomial distribution and the user is therefore for further information referred to the Binomial discussion.
Note that in the zero modified binomial distribution the relationship between the probability of “an event” and the distribution mean is different and considerably more complex than in the usual Binomial procedure. If you input the mean expected count (the average number of location where you expect to observe a laceration) the program will echo the probability of the patient being hit by a location with an event, the variance and part of the distribution. Different from the usual binomial, where the mean can be any value above zero, the mean in the zero truncated distribution can only have a value above one, the mean for the zero modified can also have a value between zero and one. The expected can also be entered as a proportion, a value between zero and one, which gives the probability for an event to occur. An assumption of the binomial is that the probability of the next event is independent and constant with regard to the number of events already seen.
The zero modified binomial where the expected number of zero events is equal to zero is known as the zero truncated binomial distribution, if the expected number of zero events is relatively high it is also known as the zero inflated binomial distribution. The usual "classical" binomial distribution is also a special case of the zero modified distribution with the proportion expected in the zero box being (1-p)sz. Check the "classical" box for this distribution.
Please study the binomial distribution further by using the Binomial spreadsheet