CS 540-1
Fall 2016
CS 540-1: Introduction to Artificial Intelligence
Homework Assignment # 4 Sample Answers
Problem 1. Probability [45]
1.
[15] Roll 2 fair 6-sided dice. What is the probability that the sum of the two dice is 7?
2.
[15] John and Robert play a game with a pile of 52 cards. The player who gets the Ace of Hearts
wins. John and Robert pick up a card each. John shows his card to Robert first, but John’s card
is not the Heart Ace. What is the probability of Robert winning?
3.
[15] 1 out of 10000 clover leaves has four leaflets. Assuming clover leaves are independent. How
many clover leaves does one need to inspect in order to find a four-leaf clover with probability at
least 0.5?
SOLUTION:
1.
1
6
There are 6 pairs whose sum is 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) where (x, y) represents
x points on the first dice and y points on the second points. The probability of having x (1 ≤ x
1
≤6) points on a dice is 61 , hence, the probability of having a particular pair (x, y) is 16 * 16 = 36
.
1
1
Therefore, the probability of having one of 6 pairs is 6 * 36 = 6 .
2.
1
51
Since John have showed his card, there are only 51 cards left. Therefore, the probability of Robert
1
having the Ace of Hearts is 51
.
3. 6932
1
The probability for a leaf clover not being 4-leaf: x = 1 - 10000
= 0.9999. Suppose we have n leaves,
so the probability for both n leaves not being 4-leaf is 0.9999n . Hence, the probability to have at
least one 4-leaf clover is 1 - 0.9999n . So, we have 1 - 0.9999n ≥ 0.5, equivalent to n > 6931.1.
Because n is a positive integer, so n = 6932.
Problem 2. Minimax Procedure [15]
Use the minimax algorithm to compute the game theoretical value at each node of the game tree below.
Problem 3. Alpha-Beta Pruning [30]
Use the tree in problem 2 to complete the questions below.
1.
2.
[15] Perform a left-to-right alpha-beta pruning on the tree. Show the alpha and beta values at
each node. Show which branches are pruned. Clearly mark the subtrees that are pruned.
[15] Do the same but from right-to-left.
CS 540-1
Fall 2016
Figure 1: The Minimax Tree
Figure 2: The left-to-right alphabeta pruning tree
CS 540-1
Fall 2016
Figure 3: The right-to-left alphabeta pruning tree