---
title: "Three urns; both balls from Urn B"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:urns-red-black-both-from-b"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/urns-red-black-both-from-b
site: BCT Engineering
---

# Three urns; both balls from Urn B

Probability and Statistics · Chapter 1: Descriptive Statistics and Basic Probability

- Asked in: 2074 Asoj, 6 marks
- Video: [Probability Question Based on Bayes' Theorem, Drawing Balls from One of Two Urns || VVI for BTech](https://www.youtube.com/watch?v=Zh6Dl6Y9nHg) — P Kalika Maths
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/urns-red-black-both-from-b
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

Write down the difference between the sample space and sample points, dependent and independent events. Urn A contains 2 white, 1 black and 3 red balls. Urn B contains 3 white, 2 black and 4 red balls. Urn C contains 4 white, 3 black and 2 red balls. One Urn is choosen at random and 2 balls are drawn. They happen to be red and black. What is the probability that both come from Urn B.

## Short answer

Using total probability and Bayes' theorem, P(both from Urn B | one red and one black) = (1/3 × 8/36) / 0.19630 = 20/53 ≈ 0.377.

## Answer

#### Sample space and sample point
- **Sample space (S):** the set of all possible outcomes of an experiment. For a die, S = {1, 2, 3, 4, 5, 6}.
- **Sample point:** a single outcome in the sample space. For a die, "3" is one sample point.

#### Dependent and independent events
- **Independent:** the occurrence of one does not change the probability of the other, so P(A ∩ B) = P(A) P(B). Example: two tosses of a coin.
- **Dependent:** the occurrence of one changes the probability of the other, so P(A ∩ B) = P(A) P(B|A). Example: drawing two balls without replacement.

#### Problem
The question is read as: given that the two balls drawn are one red and one black, find the probability that they came from Urn B.

Let R·B be the event "one red and one black". Each urn is chosen with probability 1/3.

| Urn | Contents (W, B, R) | Total | P(one red, one black) |
|---|---|---|---|
| A | 2, 1, 3 | 6 | (3 × 1) / C(6,2) = 3/15 = 0.2000 |
| B | 3, 2, 4 | 9 | (4 × 2) / C(9,2) = 8/36 = 0.2222 |
| C | 4, 3, 2 | 9 | (2 × 3) / C(9,2) = 6/36 = 0.1667 |

Total probability:

```text
P(RB) = (1/3)(0.2000 + 0.2222 + 0.1667) = 0.19630
```

Bayes' theorem:

```text
P(B | RB) = (1/3)(8/36) / P(RB)
          = 0.07407 / 0.19630
          = 20/53
          = 0.377
```

**Answer:** the probability that the two balls came from Urn B is 20/53 ≈ 0.377.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is a sample point?
   A. Set of all outcomes
   B. A single outcome in the sample space
   C. An event with no outcomes
   D. A probability
   Answer: B. The sample space contains all points. (Key point: Sample space and point)

2. What is P(one red and one black | Urn B)?
   A. (4/9)(2/9)
   B. 8/72
   C. (4x2)/C(9,2) = 8/36
   D. 4/9
   Answer: C. Favourable pairs over total pairs. (Key point: Likelihood in Urn B)

3. What is the probability of choosing each urn?
   A. 1/2
   B. 1/6
   C. 1/9
   D. 1/3
   Answer: D. One urn picked at random of 3. (Key point: Prior)

4. What is P(Urn B | red and black)?
   A. 20/53 = 0.377
   B. 1/3
   C. 8/36
   D. 0.196
   Answer: A. (1/3)(8/36)/0.1963. (Key point: Bayes result)

## Related in Probability and Statistics

- [Dependent events; defective bolt from machine B](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Abolt-factory-defective-machine-b) — Probability and Statistics, chapter 1
- [Independent events; computer errors per week](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aindependent-events-computer-errors) — Probability and Statistics, chapter 1, asked 2×
- [Independent events; regulators from B and C](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aindependent-events-voltage-regulators-b-c) — Probability and Statistics, chapter 1, asked 2×
- [Sample space, event, outcome definitions](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Asample-space-event-outcome) — Probability and Statistics, chapter 1
- [Distribution of defectives drawn from ten](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Adefective-items-three-drawn-distribution) — Probability and Statistics, chapter 2
- [Procedure: difference of two proportions, large sample](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Adifference-two-proportions-large-procedure) — Probability and Statistics, chapter 7, asked 2×

## Related in other subjects

- [Red-Black tree and its purpose](https://www.bctengineer.com/question.md?key=semester_4%2Fdata_structure_and_algorithm%3Ared-black-tree) — Data Structure and Algorithm, chapter 6, asked 2×
- [Black-box versus white-box testing](https://www.bctengineer.com/question.md?key=semester_5%2Fsoftware_engineering%3Ablack-box-vs-white-box) — Software Engineering, chapter 8
- [White box and black box testing](https://www.bctengineer.com/question.md?key=semester_5%2Finstrumentation_ii%3Awhite-black-box) — Instrumentation II, chapter 9
- [Random process; white noise PSDF and autocorrelation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Arandom-process-white-noise) — Communication Systems, chapter 9, asked 2×