Two cards are drawn from a well shuffled pack of 52 cards. Find the probability that, 13M

﻿﻿﻿﻿both the cards are of different colour

﻿﻿﻿﻿both the cards are face cards

﻿﻿﻿﻿one of the two cards is a king.

Let's solve the problem step by step.

### 1. Probability that both cards are of different color

A deck of 52 cards consists of 26 red cards (hearts and diamonds) and 26 black cards (spades and clubs).

- **Total number of ways to draw 2 cards from 52**: 
  \[
  \binom{52}{2} = \frac{52 \times 51}{2} = 1326
  \]

- **Ways to draw one red card and one black card**: 
  First, choose 1 red card from 26 red cards, and 1 black card from 26 black cards:
  \[
  \binom{26}{1} \times \binom{26}{1} = 26 \times 26 = 676
  \]

- **Probability**:
  \[
  \text{Probability} = \frac{676}{1326} = \frac{338}{663} \approx 0.5106
  \]

### 2. Probability that both cards are face cards

There are 12 face cards in a deck (3 face cards in each suit: Jack, Queen, King). So, there are 12 face cards in total.

- **Ways to draw 2 face cards from 12**: 
  \[
  \binom{12}{2} = \frac{12 \times 11}{2} = 66
  \]

- **Total number of ways to draw 2 cards from 52** (already calculated):
  \[
  \binom{52}{2} = 1326
  \]

- **Probability**:
  \[
  \text{Probability} = \frac{66}{1326} = \frac{11}{221} \approx 0.0498
  \]

### 3. Probability that one of the two cards is a king

There are 4 kings in a deck.

- **Ways to draw 1 king and 1 non-king**: 
  Choose 1 king from 4 kings, and 1 non-king from 48 non-king cards:
  \[
  \binom{4}{1} \times \binom{48}{1} = 4 \times 48 = 192
  \]

- **Total number of ways to draw 2 cards from 52** (already calculated):
  \[
  \binom{52}{2} = 1326
  \]

- **Probability**:
  \[
  \text{Probability} = \frac{192}{1326} = \frac{96}{663} \approx 0.1448
  \]

### Final answers:

1. **Probability that both cards are of different color**: \(\frac{338}{663} \approx 0.5106\)
2. **Probability that both cards are face cards**: \(\frac{11}{221} \approx 0.0498\)
3. **Probability that one of the two cards is a king**: \(\frac{96}{663} \approx 0.1448\)

Would you like more details on any part of the solution or have questions?

### Related questions:
1. What is the probability that both cards are of the same color?
2. What is the probability that neither card is a face card?
3. What is the probability of drawing exactly two kings?
4. What is the probability of drawing two red face cards?
5. How does the probability change if the cards are drawn one by one without replacement?

### Tip: 
When solving probability problems, remember to distinguish between dependent and independent events, as this affects how you calculate the probability!

Two cards are drawn from a well shuffled pack of 52 cards. Find the probability that both the cards are of different colour, both the cards are face cards, one of the two cards is a king.

Explore the probability of drawing two cards from a deck of 52 that are of different colors, both face cards, and where one is a king. This detailed solution uses combinatorics and probability theory. Ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation