Binomial Modeling and Frequency Comparisons for Repeated Card Draws

All flashcards

What does $X$ count in an experiment of 10 draws with replacement from a standard deck?

The number of diamonds drawn in the 10 draws.

What conditions make a repeated-trial experiment suitable for a binomial model?

A fixed number of independent trials, two outcomes per trial, and the same success probability on every trial.

How is the relative frequency of exactly $x$ successes calculated across repeated experiments?

Divide the number of experiments with exactly $x$ successes by the total number of completed experiments: $f_x/m$.

What are the mean and standard deviation of a binomial random variable with parameters $n$ and $p$?

The mean is $np$; the standard deviation is $\sqrt{np(1-p)}$.

How does a theoretical probability differ from a relative frequency in repeated experiments?

A theoretical probability is determined by the model; a relative frequency is calculated from observed or simulated results.

Quiz questions

1. In 10 independent draws with replacement from a standard deck, let $X$ count the diamonds. Which binomial model describes $X$?Answer: $X\sim B(10,1/4)$
2. Why does replacing and reshuffling each card keep successive draws independent?Answer: Each draw is made from the full deck, so earlier outcomes do not change later probabilities.
3. For $X\sim B(10,1/4)$, what is the standard deviation, to four decimal places?Answer: $1.3693$
4. For $X\sim B(10,1/4)$, which expression gives $P(X=x)$?Answer: $\binom{10}{x}(1/4)^x(3/4)^{10-x}$
5. If the number of repetitions increases while a probability model stays the same, what remains unchanged?Answer: The model’s theoretical probability distribution