Real-World Examples of Geometric Distribution (First Success Scenarios)

📚 Quick Study Guide

• 🎲 Geometric Distribution: Models the number of trials needed for the first success in a series of independent Bernoulli trials.
• 🔑 Key Properties: Independent trials, constant probability of success ($p$), and we're interested in the number of trials until the *first* success.
• 🧮 Probability Mass Function (PMF): $P(X = k) = (1-p)^{k-1} * p$, where $k$ is the number of trials until the first success.
• 📈 Expected Value: $E(X) = \frac{1}{p}$ (average number of trials until the first success).
• 📉 Variance: $Var(X) = \frac{1-p}{p^2}$

Practice Quiz

1. A basketball player makes free throws with a probability of 0.7. What is the probability that their first successful free throw is on their third attempt?

   1. 0.063
   2. 0.147
   3. 0.210
   4. 0.343

2. A company produces light bulbs, and 2% are defective. What is the probability that the first defective bulb is the fifth one produced?

   1. 0.0184
   2. 0.0200
   3. 0.0192
   4. 0.1000

3. A student is taking a multiple-choice quiz with 5 options per question. If they randomly guess, what is the probability that they get the first question correct on the second try?

   1. 0.16
   2. 0.64
   3. 0.8
   4. 0.2

4. A salesperson makes a sale on 1 out of every 10 calls. What is the probability that they make their first sale on the fourth call?

   1. 0.0001
   2. 0.0729
   3. 0.0081
   4. 0.0010

5. A website has a click-through rate of 5%. What is the probability that the first click occurs on the tenth visit?

   1. 0.0315
   2. 0.95
   3. 0.0000
   4. 0.5987

6. A geologist is searching for a specific mineral. The probability of finding it at any given site is 20%. What is the probability that they find the mineral for the first time at the third site?

   1. 0.008
   2. 0.128
   3. 0.040
   4. 0.64

7. A software tester finds that 3% of code submissions have critical errors. What is the probability that the first submission with a critical error is the second submission tested?

   1. 0.0009
   2. 0.0291
   3. 0.0300
   4. 0.9700