1. SHA-512 is a hash function that maps a message of any length to a digest, which is binary string of size 512. What is the total number of possible digests that could be generated from all messages?

2. How many byte values have an equal number of 1’s and 0’s?

= 70

3. When tossing a fair coin four times, how many events will form a full group of events? List all of these events in which the first toss results in a “heads.”

4. What are the odds that a randomly selected number between 1 and 100 will not have a digit 7? (For instance, 23 does not have a digit 7, but 73 does.)

Set of numbers between 1 and 100 having a digit 7= {7,17,27,37,47,57,67,70,71,72,73,74,75,76,77,78,79,87,97}

There are 19 numbers between 1 and 100 that have digit 7.There are 100 – 19 = 81 numbers that do not have digit 7.

Odds that a randomly selected number between 1 and 100 will not have a digit 7

=

5. There are 4 slots, each containing any of letters a, m, h, t. What are the odds that letters stored in these slots read the word math?

6. Consider strings of length 9 with elements being a, b or c. How many strings will contain at least 6 b’s?

possible number of a,b and c.a 0 1 2 3 0 1 2 0 1 0b 6 6 6 6 7 7 7 8 8 9c 3 2 1 0 2 1 0 1 0 0

Number of strings that contain at least 6 b’s

=

= 84 +252 + 252 + 84 + 36 + 72 + 36+ 9 +9 + 1

= 835

7. Let (4,3) and (7,11) be two points in a plane.

If all paths are equally likely, what would be the probability of choosing a path that begins with an up move and ends with an up move? (Each path is made up of moves that either go up one unit or over one unit to the right.)

(7,11) is 8 units up and 3 units to the right of (4,3).

Total number of paths = = 165

Number of paths that begins with an up move and ends with an up move

= =84

probability of choosing a path that begins with an up move and ends with an up move

=

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8. In a state lottery game, anyone who picks 6 numbers out of 36 wins $50,000, no matter how many people correctly guessed the winning numbers. What are the odds of winning in this game?

Probability of winning = =

Probability of losing =

Odds of winning = =

9. The first three characters of a car license plate number are letter characters.

a) What’s the probability of the first three characters being BAZ? b) What’s the probability of the second character being K?

c) What’s the probability that the first character is either A, B, or C?

10. A cube has all of its sides painted in red and then is cut into 216 cubes of the same size. All cubes are placed in an urn and are thoroughly mixed, so that the probability of being randomly picked from the urn is the same for all cubes. What is the probability that a randomly picked cube has at least one of its sides painted red? The table only shows one side of a big cube. The number indicates the number of faces of small cubes that are painted in red 3 2 2 2 2 32 1 1 1 1 22 1 1 1 1 22 1 1 1 1 22 1 1 1 1 23 2 2 2 2 3

From the above table, there are 16 cubes with one side painted red.There are 16 cubes with two sides painted red.There are 4 cubes with three sides painted red

The number of cubes with at least one of its sides painted red is equal to = = 152probability that a randomly picked cube has at least one of its sides painted red

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11. There are ten raffle tickets, two of which are winners. Find the probability that in a sample of 6 tickets there will be no more than one winning ticket.

Probability =

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