Theoretical Distribution Problems 🧮

Problem 1: Binomial Distribution 🪙

Ten coins are tossed. Find the probability of getting exactly 7 heads.

Solution:

• n = 10.
• Coin Toss → p = 0.5, q = 0.5.
• r = 7.

Formula: P(7) = 10C7 * (0.5)^7 * (0.5)^(10-7) P(7) = 120 * (0.5)^10 P(7) = 120 * 0.000976 = 0.1171.

Answer: 11.71%.

Problem 2: Poisson Distribution 🐟

A factory produces blades. 1 in 500 is defective. The blades are packed in packets of 10. Find the probability that a packet contains no defective blade. (Given e^-0.02 = 0.9802).

Solution:

• p = 1/500 = 0.002.
• n = 10.
• m = np = 10 * 0.002 = 0.02.

Formula: P(r) = (e^-m * m^r) / r! We want P(0). P(0) = (e^-0.02 * (0.02)^0) / 0! P(0) = (0.9802 * 1) / 1 = 0.9802.

Answer: 98.02% chance of no defect.

Problem 3: Normal Distribution 🔔

Income of a group is normally distributed with Mean = ₹20,000 and SD = ₹5,000. What percentage of people earn above ₹30,000?

Solution:

• µ = 20,000, σ = 5,000.
• X = 30,000.

Step 1: Calculate Z Z = (30,000 - 20,000) / 5,000 = 10,000 / 5,000 = +2.0.

Step 2: Find Area Area from Mean to Z=2.0 (from table) is 0.4772.

Step 3: Calculate Tail Area We want "Above 30,000" (Right tail). Total area on right = 0.5. Required Area = 0.5 - 0.4772 = 0.0228.

Answer: 2.28% of people.