Probability Problem on Gates in Subway station

Question

A subway station in a metro city has 10 gates, five for entering into the subway station, and five for exiting the subway station. The number of gates observed in each direction is observed at a particular time of a day. Assume that each outcome of the sample space is equally likely. 4Marks (a) What is the probability that at most one gate is open in each direction? (b) What is the probability that at least one gate is open in direction? (c) What is the probability that the number of gates open is the same in both direction? (d) What is the probability of the event that the total number of gates open is four?

Answer 1

Hints

There are 10 gates in total, and each outcome of the sample space is equally likely, I took the chance that a gate is open is 1/10!

No. Each gate is open or closed. Therefore, there are $2^{(10)}$ possibilities, each of which is equally likely. As a consequence of identifying the sample space as having $(1024)$ equally likely possibilities, the chance of any individual gate being open is exactly $\frac{1}{2}$.

The easiest way to attack part $a$ is to consider the 5 entrance gates and 5 exit gates separately. Examining only the entrance gates, the sample space is reduced to $(2^5) = 32.$ An examination of the corresponding row of Pascal's triangle indicates that the probability of there being at most 1 entrance gate open is

$$\frac{1 + 5}{1 + 5 + 10 + 10 + 5 + 1}.$$

More formally, if the chance of the success of an event is $p \in (0,1)$, and $q = (1-p)$, then the chance of exactly $k$ successes in $n$ bernoulli trials is

$$\binom{n}{k}p^k q^{(n-k)}.$$