What is Random Selection with Replacement? | Simple Explanation for Class 8

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What is a Random Selection with Replacement?

Grade Level:

Class 8

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Random selection with replacement means picking an item from a group, noting what it is, and then putting it back into the group before picking the next item. This ensures that every item has the same chance of being picked each time, even if it was picked before.

Simple Example

Quick Example

Imagine you have a box with 3 different coloured laddoos: one yellow, one orange, and one pink. You want to pick two laddoos. If you pick one, note its colour, put it back in the box, and then pick again, that's selection with replacement.

Worked Example

Step-by-Step

Let's say a cricket team has 4 main bowlers: A, B, C, D. The coach wants to randomly pick 2 bowlers for a special practice session, with replacement.

Step 1: List all possible bowlers: {A, B, C, D}.
---Step 2: For the first pick, any of the 4 bowlers can be chosen. Let's say A is picked.
---Step 3: Since it's 'with replacement', bowler A is put back into the group. So, the available bowlers for the second pick are still {A, B, C, D}.
---Step 4: For the second pick, any of the 4 bowlers can be chosen again. Let's say C is picked.
---Step 5: Another possible outcome could be picking B first, putting B back, and then picking B again.
---Answer: The possible pairs picked could be (A, C), (B, B), (D, A), etc. Notice that a bowler can be picked more than once.

Why It Matters

This concept is super important in fields like Data Science and AI, where computers need to pick data samples fairly to make accurate predictions. It's also used in Physics to model random events and in Economics to understand market behavior. Understanding this helps you build smarter apps and analyze complex systems.

Common Mistakes

MISTAKE: Thinking that once an item is picked, it cannot be picked again. | CORRECTION: Remember 'with replacement' means the item goes back, so it CAN be picked multiple times.

MISTAKE: Confusing 'with replacement' with 'without replacement'. | CORRECTION: 'With replacement' means the total number of items to choose from stays the same for each pick. 'Without replacement' means it decreases.

MISTAKE: Not understanding that each pick is independent when there's replacement. | CORRECTION: Because the item is put back, the outcome of a previous pick does not affect the chances of future picks.

Practice Questions

Try It Yourself

QUESTION: You have a bag with 5 numbered chits (1, 2, 3, 4, 5). If you pick a chit, note its number, and put it back, what is the probability of picking an odd number, then an even number? | ANSWER: Probability of odd = 3/5. Probability of even = 2/5. Since replacement, (3/5) * (2/5) = 6/25.

QUESTION: A lucky draw has 10 tickets. If 3 tickets are picked with replacement, how many different sequences of ticket numbers are possible? | ANSWER: For each pick, there are 10 possibilities. Since there are 3 picks with replacement, the total number of sequences is 10 * 10 * 10 = 1000.

QUESTION: A class has 25 boys and 15 girls. A student is selected randomly to answer a question, and then put back into the 'pool' of students. This is done twice. What is the probability that the first student selected is a boy AND the second student selected is also a boy? | ANSWER: Total students = 25 + 15 = 40. Probability of picking a boy first = 25/40 = 5/8. Since the boy is put back, the probability of picking a boy second is also 25/40 = 5/8. The probability of both events = (5/8) * (5/8) = 25/64.

MCQ

Quick Quiz

Which of the following scenarios describes a random selection with replacement?

Picking 5 players for a football team from a group of 15.

Drawing a card from a deck, noting it, and putting it back before drawing another.

Choosing 3 different toppings for a pizza from 8 options.

Selecting a student monitor for Monday, then a different student for Tuesday.

The Correct Answer Is:

Option B is correct because the card is put back, making it possible to draw the same card again. Options A, C, and D involve selecting distinct items or items without putting them back.

Real World Connection

In the Real World

When you use a music streaming app like JioSaavn or Spotify and it plays songs on 'shuffle' mode, it often uses a type of random selection. Sometimes, if a song is very popular or part of a small playlist, you might hear the same song again relatively quickly, especially if the shuffle algorithm allows for 'replacement' in its selection process.

Key Vocabulary

Key Terms

RANDOM: Happening or chosen without any pattern or plan. | SELECTION: The act of choosing something or someone. | REPLACEMENT: The act of putting something back in its original place. | PROBABILITY: The chance of something happening. | INDEPENDENT EVENTS: Events where the outcome of one does not affect the outcome of another.

What's Next

What to Learn Next

Now that you understand selection with replacement, your next step should be to learn about 'Random Selection WITHOUT Replacement'. This will show you how the possibilities change when items are NOT put back, which is also very common in real-world situations!