Q: Suppose you flip two coins. What is the correct sample space for this random experiment?

$$(\text{heads, heads}), (\text{heads, tails}), (\text{tails, heads}), (\text{tails, tails})$$

Q: In a random experiment, the relative frequency of an outcome is calculated as:

$$\text{Relative frequency} = \dfrac{\text{Number of successes}}{\text{Number of trials}}$$

Question 1

Which of the following best describes a random outcome according to the book definition?

Accepted Answer

An outcome where we know the possible values it can have, but not which particular value it takes.

Question 2

Suppose you flip two coins. What is the correct sample space for this random experiment?

Accepted Answer

$$(	ext{heads, heads}), (	ext{heads, tails}), (	ext{tails, heads}), (	ext{tails, tails})$$

Question 3

If you randomly select a digit between 0 and 9, what is the probability of selecting any particular digit?

Accepted Answer

$$0.1$$

Question 4

Which statement best describes the concept that 'chance has no memory'?

Accepted Answer

The probability of an outcome remains the same regardless of previous outcomes.

Question 5

A student claims that after flipping a coin and getting heads five times in a row, tails is 'due' to occur next. What statistical principle does this misconception violate?

Accepted Answer

Chance has no memory

Question 6

In a random experiment, the relative frequency of an outcome is calculated as:

Accepted Answer

$$	ext{Relative frequency} = \dfrac{	ext{Number of successes}}{	ext{Number of trials}}$$

Question 7

If you conduct 1000 independent trials of randomly picking a digit between 0 and 9, what is the expected number of times the digit '7' will be picked?

Accepted Answer

$$100$$

Question 8

Which method would be most appropriate to predict the variability in the number of times a digit is picked in repeated random trials?

Accepted Answer

Using binomial probabilities

Question 9

A random number generator is used to select digits between 0 and 9. After 82 trials, the digit '0' was picked 8 times. What is the observed relative frequency of '0'?

Accepted Answer

$$\dfrac{8}{82} \approx 0.098$$

Question 10

Which of the following is a real-world example of a random experiment?

Accepted Answer

Rolling a fair six-sided die

Question 11

If the probability of picking any digit from 0 to 9 is equal, what is the probability of picking either a '2' or a '5' in a single trial?

Accepted Answer

$$0.2$$

Question 12

In a simulation of 1000 trials, the observed frequency of digit '3' is 120. What is the relative frequency, and how does it compare to the expected probability?

Accepted Answer

Relative frequency is $$0.12$$, which is higher than the expected probability of $$0.1$$

Question 13

Which of the following statements about random number generators is true?

Accepted Answer

They produce numbers with equal probability for each possible value.

Question 14

A histogram of digits picked by people shows more 4s and fewer 0s than expected. What does this suggest about human ability to pick random numbers?

Accepted Answer

Humans are biased and do not pick numbers randomly.

Question 15

If you want to quantitatively assess whether the observed distribution of picked digits differs from the expected distribution, which approach is most appropriate?

Accepted Answer

Compare observed frequencies to expected frequencies using statistical tests.

Question 16

In a computer simulation, you run 1000 replicate experiments of picking a digit between 0 and 9, each with 82 trials. What distribution would you expect for the frequency of '0' picked in each experiment?

Accepted Answer

A binomial distribution with $$n = 82$$ and $$p = 0.1$$

Question 17

Which of the following is NOT a property of a truly random process?

Accepted Answer

Outcomes are predictable based on past results.

Question 18

A researcher wants to simulate the distribution of the frequency of '0' in 82 trials. Which tool would be most effective?

Accepted Answer

A computer simulation using R

Question 19

If the observed frequency of a digit is much higher than expected, what statistical concept helps determine if this is due to chance?

Accepted Answer

Percentile or p-value calculation

Question 20

Which of the following best describes the difference between observed and expected distributions in random experiments?

Accepted Answer

Observed distributions may differ from expected due to random variation, but should approximate expected as the number of trials increases.

Randomness & Probability: Foundations and Applications

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