With or without replacement
There are two ways to draw a sample, with or without replacement. With replacement means that once a person is selection to be in a sample, that person is placed back in the population to possibly be sampled again. Without replacement means that once an individual is sampled, that person is not placed back in the population for re-sampling.
Sampling with replacement:
Consider a population of potato sacks, each of which has either 12, 13, 14, 15, 16, 17, or 18 potatoes, and all the values are equally likely. Suppose that, in this population, there is exactly one sack with each number. So the whole population has seven sacks. If I sample two with replacement, then I first pick one (say 14). I had a 1/7 probability of choosing that one. Then I replace it. Then I pick another. Every one of them still has 1/7 probability of being chosen. And there are exactly 49 different possibilities here (assuming we distinguish between the first and second.) They are: (12,12), (12,13), (12, 14), (12,15), (12,16), (12,17), (12,18), (13,12), (13,13), (13,14), etc.
Sampling without replacement:
Consider the same population of potato sacks, each of which has either 12, 13, 14, 15, 16, 17, or 18 potatoes, and all the values are equally likely. Suppose that, in this population, there is exactly one sack with each number. So the whole population has seven sacks. If I sample two without replacement, then I first pick one (say 14). I had a 1/7 probability of choosing that one. Then I pick another. At this point, there are only six possibilities: 12, 13, 15, 16, 17, and 18. So there are only 42 different possibilities here (again assuming that we distinguish between the first and the second.) They are: (12,13), (12,14), (12,15), (12,16), (12,17), (12,18), (13,12), (13,14), (13,15), etc.
What's the Difference?
When we sample with replacement, the two sample values are independent. Practically, this means that what we get on the first one doesn't affect what we get on the second.
In sampling without replacement, the two sample values aren't independent. Practically, this means that what we got on the for the first one affects what we can get for the second one.
Consider a population of potato sacks, each of which has either 12, 13, 14, 15, 16, 17, or 18 potatoes, and all the values are equally likely. Suppose that, in this population, there is exactly one sack with each number. So the whole population has seven sacks. If I sample two with replacement, then I first pick one (say 14). I had a 1/7 probability of choosing that one. Then I replace it. Then I pick another. Every one of them still has 1/7 probability of being chosen. And there are exactly 49 different possibilities here (assuming we distinguish between the first and second.) They are: (12,12), (12,13), (12, 14), (12,15), (12,16), (12,17), (12,18), (13,12), (13,13), (13,14), etc.
Sampling without replacement:
Consider the same population of potato sacks, each of which has either 12, 13, 14, 15, 16, 17, or 18 potatoes, and all the values are equally likely. Suppose that, in this population, there is exactly one sack with each number. So the whole population has seven sacks. If I sample two without replacement, then I first pick one (say 14). I had a 1/7 probability of choosing that one. Then I pick another. At this point, there are only six possibilities: 12, 13, 15, 16, 17, and 18. So there are only 42 different possibilities here (again assuming that we distinguish between the first and the second.) They are: (12,13), (12,14), (12,15), (12,16), (12,17), (12,18), (13,12), (13,14), (13,15), etc.
What's the Difference?
When we sample with replacement, the two sample values are independent. Practically, this means that what we get on the first one doesn't affect what we get on the second.
In sampling without replacement, the two sample values aren't independent. Practically, this means that what we got on the for the first one affects what we can get for the second one.
List elements of a set
Conducting two-step chance experiments
A tree-diagram is a very helpful way to visualise all of the possible outcomes.
Independent Event
Independent events are events that are not affected by previous events.
For example, if you flip a coin, it has a probability of 0.5 for landing heads.To get the probability of landing 2 heads, you just multiply the probability of landing heads twice. (0.5 x 0.5) = 0.25 or 1/4.To get the probability of landing 3 heads, you just multiply the probability of landing heads three times. (0.5 x 0.5 x 0.5) = 0.125 or 1/8.
Dependant Event
Dependent events are events that are affected by previous events.
For example, if you take a marble out of a bag, the probability changes based on the results of the first result.The probability of drawing a blue marble in the first instance is 2 out of 5 or (0.4). Now there can be two possible outcomes, you could have drawn a red marble, which would then give you a 2 in 4 chance of drawing a blue marble in the second instance. Or you could have drawn a blue marble in the first instance which would leave you with a 1 in 4 chance of drawing a blue marble in the second instance.
What are the chances of drawing 2 blue marbles?
There’s a 2/5 chance (0.4) in the first instance, and a 1/4 (0.25) in the second instance. Once again we multiply the probabilities 1/10 (0.1) or 10% chance of drawing 2 blue marbles.
There’s a 2/5 chance (0.4) in the first instance, and a 1/4 (0.25) in the second instance. Once again we multiply the probabilities 1/10 (0.1) or 10% chance of drawing 2 blue marbles.