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Mathematics

Probability

Discrete Random Variables

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Drawing one card from a standard 52-card deck and getting a queen.

PMF: $P(X = x) = \begin{cases} \frac{4}{52} & \text{for } x=1 \\ \frac{48}{52} & \text{for } x=0 \end{cases}$

Counting the number of heads in 2 coin flips with a fair coin.

PMF: $P(X = x) = \begin{cases} \frac{1}{4} & \text{for } x=0 \\ \frac{1}{2} & \text{for } x=1 \\ \frac{1}{4} & \text{for } x=2 \end{cases}$

Tossing a fair six-sided die and noting the outcome.

PMF: $P(X = x) = \frac{1}{6}$ for $x = 1, 2, 3, 4, 5, 6$

A basketball player makes a free throw shot with a 70% success rate.

PMF: $P(X = x) = \begin{cases} 0.70 & \text{for } x=1 \\ 0.30 & \text{for } x=0 \end{cases}$

Flipping 3 fair coins and counting the number of tails.

PMF: $P(X = x) = \begin{cases} \frac{1}{8} & \text{for } x=0 \\ \frac{3}{8} & \text{for } x=1 \\ \frac{3}{8} & \text{for } x=2 \\ \frac{1}{8} & \text{for } x=3 \end{cases}$

Picking a random day of the week.

PMF: $P(X = x) = \frac{1}{7}$ for $x$ being each day of the week

Drawing a single card from a deck and it being an ace or not.

PMF: $P(X = x) = \begin{cases} \frac{4}{52} & \text{for } x=1 \\ \frac{48}{52} & \text{for } x=0 \end{cases}$

Observing the outcome of a single traffic light (red, yellow, green) assuming each outcome is equally likely.

PMF: $P(X = x) = \frac{1}{3}$ for $x =$ red, yellow, or green

Rolling two dice and counting the number of sixes.

PMF: $P(X = x) = \begin{cases} \frac{25}{36} & \text{for } x=0 \\ \frac{10}{36} & \text{for } x=1 \\ \frac{1}{36} & \text{for } x=2 \end{cases}$

Counting the number of a particular book sold in a bookstore in one day, given past data shows they sell 0, 1, or 2 copies with probabilities of 60%, 30%, and 10% respectively.

PMF: $P(X = x) = \begin{cases} 0.60 & \text{for } x=0 \\ 0.30 & \text{for } x=1 \\ 0.10 & \text{for } x=2 \end{cases}$

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