4. Discrete Probability Distributions

1. Binomial Distribution

• Use: Fixed number of trials ($n$), two outcomes (Success/Failure), independent trials.
• Example (Quality Control):
• A factory produces bulbs. 10% are defective. You pick 5 bulbs ($n=5$).
• Question: What is the probability exactly 2 are defective ($k=2$)?
• $p = 0.1$ (prob of defect), $1-p = 0.9$.
• Calculation:

$$ P(X=2) = \\binom{5}{2} (0.1)^2 (0.9)^3 $$ $$ P(X=2) = 10 \\times 0.01 \\times 0.729 = 0.0729 $$ (There is a 7.29% chance of finding exactly 2 defective bulbs).

2. Geometric Distribution

• Use: Repeating trials until the first success occurs.
• Example (Basketball):
• A player makes a free throw 20% of the time ($p=0.2$).
• Question: What is the probability he makes his first shot on the 3rd attempt?
• This means: Miss, Miss, Make.
• Calculation:

$$ P(X=3) = (1-p)^{3-1} \\times p $$ $$ P(X=3) = (0.8)^2 \\times 0.2 = 0.64 \\times 0.2 = 0.128 $$

3. Poisson Distribution

• Use: Counting events in a fixed interval (time/area) given an average rate (\\lambda).
• Example (Call Center):
• A call center receives an average of 4 calls per minute (\\lambda = 4).
• Question: What is the probability of receiving exactly 6 calls in a minute?
• Calculation:

$$ P(X=6) = \frac{e^{-4} \cdot 4^6}{6!} $$ $$ P(X=6) = \frac{0.0183 \cdot 4096}{720} \approx 0.104 $$ (There is a 10.4% chance of getting exactly 6 calls).