# Verifying answer to Probability and Random variable question

Two friends communicate through messages in a bottle. They send these bottles through a channel. Probability of bottle reaching from Friend1 to 2 is $$\frac {5}{6}$$.

And the probability of receiving a bottle is $$\frac {1}{3}$$ (Friend1 receives a confirmation bottle from friend 2).

However, it is not necessary that when a friend1 sends a bottle it reaches friend2. And if friend1 doesn't receive a confirmation bottle within some time then he sends the bottle again. X is a random variable that represents the number of attempts made by friend1 to communicate a message successfully to friend2.

Find probability that X is 2.

In order to solve this question I used geometric distribution. Here is how I have done it

$$P(X=2) = (1- \frac {5}{6}) \cdot \frac {5}{6}$$

Is this correct?

• Where/how does the 1/3 probability of receiving come into the picture? Oct 1, 2018 at 17:17
• I don't think this is clear. What does the $\frac 13$ represent?
– lulu
Oct 1, 2018 at 17:18
• @lulu Just edited. I hope it's clear now. Oct 1, 2018 at 17:20
• Not really. I think the bit about the bottles is just confusing, as in the phrase "And if friend1 doesn't receive a bottle within some time then he sends the bottle again." How can he send it if he never received it? Can you phrase the problem without referring to bottles in anyway?
– lulu
Oct 1, 2018 at 17:21
• Is confirmation required for the communication to be considered "successful"? If no confirmation is required, then you are correct. But, if the confirmation bottle does not arrive successfully, keep in mind that Friend 1 will continue sending bottles until it is received. Oct 1, 2018 at 17:23

Assuming that $$\frac {1}{3}$$ is a probability that bottle sent by friend2 is successfully recieved by friend1; though it is useless here.

Friend1 wants to send a bottle to friend2, such that: friend2 must recieve the bottle.
You have been given X=2, i.e. number of attempts in which friend1 communicates to friend2 is 2.

So, definitely, first attempt must be a failure.
(friend1 may not communicate again if first bottle successfully reached friend2, why?)

Probability of failure = 1 - probability of success:
$$1- \frac {5}{6} = \frac {1}{6}$$ Now, second attempt must be a success, so that friend1 communicates.
Here probability is simply $$\frac {5}{6}$$

Total probability is product of these 2 probabilities:
$$\frac {1}{6} \cdot \frac {5}{6} = \frac {5}{36}$$