Picking fairly | Probability and combinatorics | Probability and Statistics | Khan Academy

Summary

In this engaging video, a magician at a birthday party explores three methods for selecting a volunteer from 15 kids, evaluating their fairness. Method 1, involving 100 pieces of paper, results in unequal chances as some kids receive more papers than others. Method 2, with 75 pieces, ensures each child gets an equal share, making it fair. Method 3 distributes 30 pieces but uses the uneven count of windows in the room for selection, leading to randomness in results. Ultimately, the video illustrates the importance of fair methods in random selection.

Takeaways

• 😀 The magician aims to select a volunteer fairly from a circle of 15 kids.
• 🤔 Fairness means each child should have the same chance of being chosen.
• 📝 In Method One, 100 pieces of paper are distributed, but not all kids receive the same number.
• ⚖️ Since 100 is not divisible by 15, some kids get 6 pieces, while others get 7, resulting in unequal chances.
• ✅ Method One is deemed **Not Fair** due to the unequal distribution of paper.
• 📄 Method Two involves distributing 75 pieces of paper, allowing for even distribution.
• 🔢 Because 75 is divisible by 15, each child receives exactly 5 pieces of paper.
• 👍 Method Two is considered **Fair** since all kids have an equal chance of being selected.
• 🪟 In Method Three, 30 pieces of paper are passed out evenly, with each child getting 2 pieces.
• ❌ However, using the number of windows in the room as a selection method is **Not Fair** because window counts are not evenly distributed.

Q & A

What is the purpose of the magician's methods in the birthday party scenario?

The magician wants to choose a volunteer from a circle of 15 kids, ensuring that each child has the same chance of being selected.

How does Method One distribute the pieces of paper?

In Method One, the magician hands out 100 pieces of paper numbered 1 to 100 in a clockwise manner until all pieces are distributed.

Why is Method One considered unfair?

Method One is unfair because 100 is not divisible by 15, leading to unequal distribution of pieces of paper among the kids. Some kids receive 7 pieces while others receive only 6.

What does the magician do after distributing the pieces of paper in Method One?

After distributing the pieces of paper, the magician uses a random number generator to pick an integer between 1 and 100 to select the volunteer.

How is Method Two different from Method One?

Method Two uses 75 pieces of paper instead of 100, and since 75 is divisible by 15, each child receives an equal number of pieces.

What makes Method Two fair?

Method Two is fair because each of the 15 kids receives 5 pieces of paper, giving them all an equal chance of being selected by the random number generator.

How does the magician select a volunteer in Method Three?

In Method Three, the magician distributes 30 pieces of paper numbered 1 to 30, ensuring each child gets 2 pieces. The volunteer is then chosen based on the number of windows in the room.

Why is Method Three deemed unfair?

Method Three is unfair because the number of windows in different houses is not evenly distributed, making it an unreliable method for random selection.

What is a key takeaway regarding fairness from the magician's methods?

The key takeaway is that fairness in selection methods relies on equal distribution of chances; both Methods One and Three fail to provide this, while Method Two successfully does.

What general principle can be derived about random selection from the magician's example?

The example illustrates that for a random selection process to be fair, all participants must have an equal likelihood of being chosen, which can be affected by how items (like pieces of paper) are distributed.