0.8: Exercise

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Exercise 0.1.

Rewrite each of these expressions as something of the form \(2^{x}\).

\(\left(2^{n}\right)^{n}=? ?\)
\(2^{n}+2^{n}=? ?\)
\(\left(2^{n}\right)\left(2^{n}\right)=? ?\)
\(\left(2^{n}\right) / 2=? ?\)
\(\sqrt{2^{n}}=? ?\)
\(\left(2^{n}\right)^{2}=? ?\)

Exercise 0.2.

(a) What is \(0+1+2+\cdots+(n-2)+(n-1) \% n\), when \(n\) is an odd integer? Prove your answer!

(b) What is \(0+1+2+\cdots+(n-2)+(n-1) \% n\), when \(n\) is even? Prove your answer!

Exercise 0.3.

What is \((-99) \% 10\) ?

Exercise 0.4.

Without using a calculator, what are the last two digits of \(357998^{6} ?\)

Exercise 0.5.

Without using a calculator, what is 1000 ! % 427 ? (That’s not me being excited about the number one thousand, it’s one thousand factorial!)

Exercise 0.6.

Which values \(x \in \mathbb{Z}_{11}\) satisfy \(x^{2} \equiv_{11} 5\) ? Which satisfy \(x^{2} \equiv_{11} 6\) ?

Exercise 0.7.

What is the result of XOR’ing every \(n\) bit string? For example, the expression below is the XOR of every 5-bit string:

\[00000 \oplus 00001 \oplus 00010 \oplus 00011 \oplus \cdots \oplus 11110 \oplus 11111 \notag\]

Give a convincing justification of your answer.

Exercise 0.8.

Consider rolling several \(d\)-sided dice, where the sides are labeled \(\{0, \ldots, d-1\}\).

(a) When rolling two of these dice, what is the probability of rolling snake-eyes (a pair of 1s)?

(b) When rolling two of these dice, what is the probability that they don’t match?

(d) When rolling three of these dice, what is the probability that they don’t all match (including the case where two match)?

(e) When rolling three of these dice, what is the probability that at least two of them match (including the case where all three match)?

(f) When rolling three of these dice, what is the probability of seeing at least one 0 ?

Exercise 0.9.

When rolling two 6-sided dice, there is some probability of rolling snake-eyes (two \(1 \mathrm{~s}\) ). You determined this probability in the previous problem. In some game, I roll both dice each time it is my turn. What is the smallest value \(t\) such that:

\(\operatorname{Pr}[\) I have rolled snake-eyes in at least one of my first \(t\) turns \(] \geqslant 0.5 ?\)

In other words, how many turns until my probability of getting snake-eyes exceeds \(50 \%\) ?