Understanding the Probability of Random Samples from a Bag of Coins

A bag contains 10p, 20p, and 50p coins in the ratio 1:2:2. If we take a random sample of 3 coins, what are the possible samples that can be obtained?

Analysis of the Ratio and Distribution

First, let's analyze the given distribution. If the number of 10p coins is x, then the number of 20p coins is 2x, and the number of 50p coins is also 2x. This means the total number of coins in the bag is:

x 2x 2x 5x

Possible Samples without Replacement

For a random sample of 3 coins without replacement, the possible combinations can be:
- All 3 coins of the same type - 2 coins of one type and 1 coin of another type - 1 coin of each type

Combinations of 3 Coins of the Same Type

3 x 10p coins: 10p 10p 10p 3 x 20p coins: 20p 20p 20p 3 x 50p coins: 50p 50p 50p

Combinations of 2 Coins of One Type and 1 Coin of Another Type

2 x 10p and 1 x 20p: 10p 10p 20p 2 x 10p and 1 x 50p: 10p 10p 50p 2 x 20p and 1 x 10p: 20p 20p 10p 2 x 20p and 1 x 50p: 20p 20p 50p 2 x 50p and 1 x 10p: 50p 50p 10p 2 x 50p and 1 x 20p: 50p 50p 20p

Combinations of 1 Coin of Each Type

1 x 10p, 1 x 20p, and 1 x 50p: 10p 20p 50p

Summary of Possible Samples

Putting these combinations together, the possible samples of 3 coins are:

10p 10p 10p 20p 20p 20p 50p 50p 50p 10p 10p 20p 10p 10p 50p 20p 20p 10p 20p 20p 50p 50p 50p 10p 50p 50p 20p 10p 20p 50p

Considerations for Random Sample Selection

The term "random sample" can be ambiguous in this context. It can refer to either:

A sample taken without replacement, where each coin is pulled out one by one, and each coin remains in the bag after being pulled out. A sample taken with replacement, where each coin is pulled out, noted, and then returned to the bag, and the process is repeated three times.

Additionally, whether the order in which the coins are drawn is significant also adds to the complexity.

Without Replacement Sample

Without replacement, the number of possible samples depends on the total number of coins in the bag:

If there are 5 coins, the possible samples are as follows: 10/20/20, 10/50/50, 20/20/50, 20/50/50, 10/20/50 If there are 10 coins, in addition to the possibilities above, new combinations like 10/10/20, 10/10/50, 20/20/20, and 50/50/50 also appear. With 15 or more coins, the additional combinations 10/10/10 are added.

Therefore, the number of possible samples without considering order could be 5, 9, or 10, depending on the total number of coins in the bag.

With Replacement Sample

With replacement, the number of possible samples is always higher:

Without considering the order, the number of samples remains at 10. Considering the order, the number of samples increases to 18, 26, or 27, depending on the total number of coins in the bag.

Ensuring Correct and Sufficient Samples

To ensure there are enough and correct samples, several steps can be taken:

Define the Sample Size and Distribution: Ensure the total number of coins in the bag is known and accurately reflects the given ratio. Sampling Method: Decide whether to sample without or with replacement based on the specific requirements of the problem. Order Consideration: Determine whether the order in which the coins are drawn is significant, and adjust the number of possible outcomes accordingly. Verification: Conduct pilot tests or simulations to ensure the sampling method yields representative results. Documentation: Document the sampling process, including the method, number of coins, and conditions, to provide transparency and reproducibility.