1. A binary tree has ten nodes. The inorder and preorder traversals of the binary tree are shown as follows.
 Inorder: FKCGJBDEAI
 Preorder: GFCKDJBIEA
Draw the tree.

a. Determine the entropy of the data set. (4 points)

b. Determine the average entropy of the resulting subsets obtained by splitting on attribute A. (4 points)

c. How much information gain is obtained if attribute A is selected for the root node of the decision tree? (2 points) Note: Please use the values: log23 = 1.6 and log25 = 2.3. Hint: Entropy = is distribution for event i

3. To transmit text across a network, an English character is usually encoded in 8 bits in which bits 0~6 are the ASCII code and bit 7 is used for parity checking. Suppose a string of 5 characters, infni, is to be transmitted and odd-parity checking is adopted. Let the ASCII code of i be 1101001, the ASCII code of n be 1101110, and the ASCII code of f be 1100110. What is the sequence of 40 bits encoded for the 5-character string? Please show the sequence in hexadecimal (base 16) form.

4. Suppose we have an array A containing 9 integers: 240, 119, 682, 341, 225, 556, 122, 661, 245, i.e. A = [240, 119, 682, 341, 225, 556, 122, 661, 245] with the first index being 0. Please convert the array A into a maxheap and show the content of A after conversion. Note that the conversion should be done in-place.

（一）丟擲第一次後雙方身上可能各有幾枚代幣？【4 分；未寫出推論過程者不予計分】

（二）丟擲第二次後雙方身上可能各有幾枚代幣？【4 分；未寫出推論過程者不予計分】

（三）會在丟擲第三次後就停止遊戲的機率是多少？【4 分；未寫出推論過程者不予計分】

（四）會在丟擲第四次後就停止遊戲的機率是多少？【4 分；未寫出推論過程者不予計分】

（五）會在丟擲第五次後就停止遊戲的機率是多少？【4 分；未寫出推論過程者不予計分】

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