i have to solve this excercise, but i'm confussed how to find the probability mass function and if it is an binomial distribution with a bernoulli trials. A mouse is placed in a model with five doors. Each door opens when the mouse is detected by a sensor, this has a probability 0.7; the operation of the doors is independent of each other. The following figure shows the basic outline of the model that the mouse will enter. e= entry s=sensor p=door l= arrival
If the random variable is: X: = "The number of direct paths from the sensor to arrival available on the model" direct path: in which there is no need to travel a section of the route more than once.
the question is what is the probability mass function of the problem?
Essentially, a direct path is one where one the first detection of the sensor, the appropriate doors open up to create a straight path from $S$ to $L$. Hence
$$P(X=0) : \text{No direct paths}$$ $$P(X=1) : \text{Exactly one direct path}\\.\\.\\.\\P(X=4) : \text{Exactly 4 direct paths}$$
Now, you just consider each case separately. For example, for $X=0$, which gates need to be shut, and which can be open? Can you proceed?