Probability Problems 🧮

Problem 1: Odds in Favor/Against 🎲

If P(A) = 2/5, find the Odds in Favor and Odds Against.

Solution to Problem 1 💡

• P(A) = 2/5 (Successes = 2, Total = 5).

• Failures = Total - Success = 5 - 2 = 3.

• Odds in Favor (Success : Failure): 2 : 3

• Odds Against (Failure : Success): 3 : 2

Problem 2: At Least One 🎯

A problem is given to 3 students A, B, C. Their chances of solving it are 1/2, 1/3, 1/4 respectively. What is the probability that the problem will be solved?

Solution to Problem 2 💡

"Problem is solved" means At Least One student solves it. It's easier to use: 1 - P(None Solved).

• P(A) = 1/2 → P(A') = 1/2
• P(B) = 1/3 → P(B') = 2/3
• P(C) = 1/4 → P(C') = 3/4

P(None) = P(A') * P(B') * P(C') (Independent) 1/2 * 2/3 * 3/4 = 6/24 = 1/4

P(At Least One) = 1 - 1/4 = 3/4.

Answer: 75% chance.

Problem 3: Cards 🃏

From a pack of 52 cards, two cards are drawn at random. Find probability that one is a King and other is a Queen.

Solution to Problem 3 💡

• Total cards = 52. Selection = 2.

• Total Cases (n) = 52C2 = (52*51)/(2*1) = 1326.

• Favorable Cases (m):

  • One King from 4 Kings: 4C1 = 4.
  • One Queen from 4 Queens: 4C1 = 4.
  • Total ways = 4 * 4 = 16.

• Probability = m/n = 16 / 1326 = 8 / 663.

Problem 4: Balls & Urns 🔴🔵

Urn A: 3 Red, 4 White. Urn B: 5 Red, 3 White. One ball is drawn from each. Find probability that both are Red.

Solution to Problem 4 💡

Events are Independent (Separate Urns).

• P(Red from A): 3/7.
• P(Red from B): 5/8.

P(Both Red) = 3/7 * 5/8 = 15/56.