Given N which denotes the initial position of the person on the number line. Also given L which is the probability of the person of going left. Find the probability of reaching all points on the number line after N moves from point N. Each move can be either to the left or to the right.
Input: n = 2, l = 0.5
Output: 0.2500 0.0000 0.5000 0.0000 0.2500
The person can’t reach n-1th position and n+1th position in 2 passes, hence the probability is 0. The person can reach 0th position by only moving 2 steps left from index 2, hence the probability of reaching 0th index is 05*0.5=0.25. Similarly for 2n index, the probability is 0.25.
Input: n = 3, l = 0.1
Output: 0.0010 0.0000 0.0270 0.0000 0.2430 0.0000 0.7290
The person can reach n-1th in three ways, i.e., (llr, lrl, rll) where l denotes left and r denotes right. Hence the probability of n-1th index is 0.027. Similarly probabilties for all other points are also calculated.
Approach: Construct an array arr[n+1][2n+1] where each row represents a pass and the columns represent the points on the line. The maximum a person can move from index N is to 0th index at left or to 2nth index at right. Initially the probabilities after one pass will be left for arr[n-1] and right for arr[n+1]. The n-1 moves which are left will be done, hence the two possible moves will either be n steps to the right or n steps to the left. So the recurrence relations for right and left moves for all will be:
arr[i][j] += (arr[i – 1][j – 1] * right)
arr[i][j] += (arr[i – 1][j + 1] * left)
The summation of probabilities for all possible moves for any index will be stored in arr[n][i].
Below is the implementation of the above approach:
0.2500 0.0000 0.5000 0.0000 0.2500
Time Complexity: O(N2)
Auxiliary Space: O(N2)
- Probability of reaching a point with 2 or 3 steps at a time
- Find maximum path sum in a 2D matrix when exactly two left moves are allowed
- Puzzle | Find The Probability of Distance in a Square
- Find the probability of a state at a given time in a Markov chain | Set 1
- Find probability of selecting element from kth column after N iterations
- Find probability that a player wins when probabilities of hitting the target are given
- Find number of endless points
- Find maximum points which can be obtained by deleting elements from array
- Minimum Cost Path with Left, Right, Bottom and Up moves allowed
- Probability of rain on N+1th day
- Aptitude | Probability | Question 5
- Aptitude | Probability | Question 6
- Aptitude | Probability | Question 7
- Aptitude | Probability | Question 8
- Probability that two persons will meet
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.