Practice Randomness in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Randomness is the quality of having no predictable pattern at the individual level, yet following precise probability rules over many repetitions โ€” outcomes are uncertain one at a time but statistically regular in the long run.

Truly random means you can't predict the next outcome, even with complete information.

Showing a random 20 of 50 problems.

Example 1

medium
A random shuffle of 5 distinct cards produces the original sorted order. Is this evidence the shuffle is broken?

Example 2

challenge
A fair coin is flipped 3 times. What is the probability of getting at least one head, and why is this not simply 3ร—123\times\frac{1}{2}?

Example 3

hard
Six fair coins are flipped. What is the probability of exactly 3 heads?

Example 4

challenge
A pseudo-random generator passes every statistical test you apply, yet its entire output is determined by a hidden seed. Is its output truly random? Explain the distinction.

Example 5

easy
A fair die is rolled. After three 6's in a row, what is the probability the next roll is also a 6?

Example 6

easy
Over many fair coin flips, the proportion of heads approaches what value?

Example 7

hard
In a sample of 1000 fair coin flips, the law of large numbers says the proportion of heads is close to 0.5. About how many heads do you expect?

Example 8

challenge
In 10 fair coin flips, is the most likely single specific sequence (each has probability 1/10241/1024) more or less likely than getting exactly 5 heads (in any order)?

Example 9

medium
A scientist sees a 'cluster' of 3 disease cases on one street and suspects a cause. Why might pure randomness explain this?

Example 10

medium
A coin is flipped 4 times. How many possible ordered sequences are there?

Example 11

easy
Is rolling a fair die a random process at the level of a single roll?

Example 12

easy
After 4 heads in a row with a fair coin, what is the probability the next flip is heads?

Example 13

medium
Why does a casino profit reliably from random games even though each individual spin is unpredictable?

Example 14

easy
Is the digit sequence of ฯ€\pi (3,1,4,1,5,...) random in the everyday sense of unpredictable to compute?

Example 15

medium
Why is the lottery sequence 17, 22, 31, 38, 41, 46 typically considered 'more random-looking' than 1, 2, 3, 4, 5, 6 even though both have the same probability?

Example 16

medium
A fair coin is flipped 4 times. How many equally likely outcome sequences are there, and what is the probability of HHHH?

Example 17

medium
Explain why 'random' is not the same as 'each outcome happens equally often in every short run.'

Example 18

easy
A student says 'I picked lottery numbers 1,2,3,4,5,6 โ€” these can't win because they're not random.' Evaluate this claim.

Example 19

easy
True or false: in true randomness, you cannot predict the next single outcome even with complete information.

Example 20

easy
Identify the random process: (a) the result of 343^4, (b) the next song shuffled on a playlist set to random.
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