Example: Combinatorics and probability | Probability and combinatorics | Precalculus | Khan Academy

TL;DR

Analyzing the probability of getting all four 1's in a hand of 9 cards when selecting from a deck of 36 unique cards.

Transcript

A card game using 36 unique cards, four suits, diamonds, hearts, clubs, and spades, with cards numbered from 1 to 9 in each suit. So there's four suits. Each of them have nine cards, so that gives us 36 unique cards. A hand is a collection of nine cards, which can be sorted however the player chooses. So they're essentially telling us that order do... Read More

Key Insights

• #️⃣ The probability of an event is the number of ways it can happen divided by the total number of possibilities.
• 🤗 The total number of possible hands is calculated using combinations, which accounts for the order not mattering.
• 🤗 The number of hands with four 1's is determined by considering the remaining five cards that can be selected from the deck.
• 🤗 Dividing the number of hands with four 1's by the total number of possible hands gives the probability of getting all four 1's in a hand.

Questions & Answers

Q: What is the probability of getting all four 1's in a hand of nine cards from a deck of 36 unique cards?

The probability is approximately 2 in 935, which means there is roughly a 1 in 500 chance of getting all four 1's in a hand of nine cards.

Q: How many different hands can be formed from the deck of 36 unique cards?

The total number of possible hands is calculated using combinations and is equal to 36 choose 9, which simplifies to 36! / (36-9)! * 9!.

Q: How are the different arrangements of the cards taken into account in the probability calculation?

To avoid overcounting for the different orderings of the cards, the total number of hands is divided by the number of ways nine cards can be arranged, which is calculated as 9!.

Q: Why is the number of ways to have four 1's in a hand directly related to the remaining five cards in the hand?

To have four 1's in a hand, the remaining five cards can be selected from the remaining 32 cards in the deck. The number of ways to arrange these five cards is then divided to avoid counting the same combination multiple times.

Summary & Key Takeaways

• A deck of 36 unique cards consists of four suits with nine cards in each suit.

• A hand of nine cards can be arranged in any order, and the order does not matter.

• The probability of getting all four 1's in a hand of nine cards from the deck of 36 unique cards is approximately 2 in 935.