How are probability problems solved?

Question

Mathematics

(The picture is clickable)
Can you tell me the algorithm for solving such problems?
And what to read on this topic?

Answer 1

Specifically for this task, the algorithm is simple. At each step:
Winning (leaving the forest) - one road out of three:
W = 1/3
Losing (robbers) - two roads out of three * probability of meeting robbers:
L = 2p/3
Left in the game - two roads out of three * probability do not meet robbers:
G = 2(1-p)/3
Accordingly, for the i - th step:
W _i = 1/3*G ^{( i -1)}
L _i = 2p/3*G ^{( i -1)}
Ratio
W _i /L _i = (1/3)/(2p/3) = 1/(2p)
does not depend on i , and hence for
SUM_{i =1..N} (W _i )/SUM _{i =0..N} (L _i ) = 1/(2p)
for any N
Thus, the total probability of winning
P _W = W/(W+L) = 1/ (1+2p)
Well, read - probability theory and combinatorics.

Answer 2

Probability books, obviously.

Answer 3

Specifically, this problem can be solved as follows:
And so, we have p - the probability of an attack by robbers, 0 <= p <= 1. Let q be the probability of leaving the forest.
Let's consider what variants of events can happen and with what probability:
1. With probability 0,(3) the hero leaves the forest.
2. Robbers attack with probability 0,(6)*p.
3. With probability 0.(6)*(1-p) the hero goes to another city. (recursive variant)
Since variant 3 is recursive, it does not change the ratio of the probabilities of getting_out_of_forest and attacks_of_robbers. This means that in the end the ratio will be 0, (3) to 0, (6) * p, and their sum will be equal to one.
We get the equation:
q / ( 1 - q ) = 0.(3) / ( 0.(6) * p )
We solve, we get the following expression:
q = 0.(3) / ( 0.(6)*p + 0.(3) )

Answer 4

Look at Markov chains. On Wikipedia, the English article is quite suitable: https://en.wikipedia.org/wiki/Markov_chain
Here, of course, everything is simpler. But if the cities were connected more cunningly, and not from all of them there were roads to salvation, then the solution would be behind the chains.