a man dealt four spade cards from an ordinary pack of 52 cards if is given three more card find the probability that at least one of the additional card is also a spade

Let's solve the problem step by step.

### Problem Breakdown:
- **Initial Condition**: A man is dealt four spade cards from an ordinary deck of 52 cards.
- **New Condition**: He is given three more cards.
- **Objective**: Find the probability that at least one of the additional cards is also a spade.

### Step 1: Understand the Composition of the Deck
Initially, 4 spades have been dealt out of the 13 spades in the deck. So, there are:
- 48 cards remaining in the deck (since 4 cards have already been dealt).
- 9 spades remaining in the deck (since 4 spades have already been dealt).

### Step 2: Calculate the Probability of No Spades in the Additional Three Cards
To find the probability of getting at least one spade, it is easier to first find the probability of getting no spades and then subtract that from 1.

The probability that the first additional card is not a spade is:
\[
P(\text{no spade on 1st card}) = \frac{48 - 9}{48} = \frac{39}{48}
\]

The probability that the second additional card is not a spade (given that the first wasn't) is:
\[
P(\text{no spade on 2nd card}) = \frac{38}{47}
\]

The probability that the third additional card is not a spade (given that the first two weren't) is:
\[
P(\text{no spade on 3rd card}) = \frac{37}{46}
\]

### Step 3: Calculate the Combined Probability of No Spades in All Three Cards
The probability of getting no spades in all three cards is:
\[
P(\text{no spade in all 3 cards}) = \frac{39}{48} \times \frac{38}{47} \times \frac{37}{46}
\]

### Step 4: Calculate the Probability of Getting at Least One Spade
The probability of getting at least one spade is the complement of getting no spades:
\[
P(\text{at least 1 spade}) = 1 - P(\text{no spade in all 3 cards})
\]

Let's compute this probability.The probability that at least one of the additional three cards is a spade is \(\frac{8157}{17296}\), which is approximately 0.472 (or 47.16%).

Would you like more details or have any questions?

Here are five related questions to explore further:
1. What is the probability of getting exactly one spade among the three additional cards?
2. How would the probability change if the number of additional cards were increased to four?
3. What is the probability of getting no spades at all in the initial four cards dealt?
4. How would the probability differ if the initial cards dealt were from a deck with jokers?
5. Can you calculate the probability of getting exactly two spades in the additional three cards?

**Tip**: Understanding how to calculate complementary probabilities (like "at least one") can simplify many probability problems.

A man is dealt four spade cards from an ordinary pack of 52 cards. If he is given three more cards, find the probability that at least one of the additional cards is also a spade.

Explore how to calculate the probability of drawing at least one spade when three additional cards are dealt from a standard deck. Learn key probability concepts, including complementary probability and combinatorics. Suitable for students in Grades 10-12.

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