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