A student is conducting an experiment in which he randomly selects a card from a standard 52-card deck, records if the card is a heart or not, puts the card back in the deck, and repeats 5 more times (i.e., he pulls a total of 6 cards in each experiment).
- What are the values of the parameters n and p?
- What is the mean number of hearts pulled across many 6 trial experiments? Show all work.
- What is the standard deviation of the number of hearts drawn across many 6 trial experiments?
Solution
Let $X$ denote the number of heart card drawn from a deck of 52 cards in 6 trials.
The probability of getting heart card in a single draw of a card from a deck of 52 card is $p = \frac{13}{52}=\frac{1}{4}=0.25$.
The trials are independent of each other.
- The random variable $X$ follows Binomial distribution with parameters $n = 6$ and $p=0.25$.
Thus $X\sim B(6, 0.25)$.
The probability mass function of $X$ is
$$ \begin{aligned} P(X=x) &= \binom{6}{x} (0.25)^x (1-0.25)^{6-x},\\ &\quad x=0,1,\cdots, 6 \end{aligned} $$
- The mean number of hearts pulled across many 6 trial experiments is
$$ \begin{aligned} \mu&=E(X)\\ &=n*p \\ &= 6 \times 0.25\\ &= 1.5 \end{aligned} $$
- The standard deviation of the number of hearts drawn across many 6 trial experiments is
$$ \begin{aligned} \sigma&= \sqrt{n*p*(1-p)}\\ &= \sqrt{6 \times 0.25 \times (1- 0.25)}\\ &=1.0607 \end{aligned} $$
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
