a) What is sampling with replacement? (b) What is sampling without replacement? c) How does sampling without replacement affect the probability of events, compared to sampling with replacement?
Sampling with replacement puts the chosen unit back in the pool, while without replacement does not.
Probabilities remain the same with replacement, but both numerator and denominator are reduced by one in without replacement in subsequent choices.
a) Sampling with replacement refers to the process of selecting an item from a population and then putting it back before selecting the next item. This means that each item has an equal chance of being selected in each draw, regardless of whether it has already been selected before.
b) Sampling without replacement, on the other hand, involves selecting an item from a population but not replacing it before the next selection. In this case, the probability of selecting each subsequent item changes depending on the previous selections.
c) Sampling without replacement affects the probability of events by reducing the likelihood of certain outcomes as the sampling progresses. As items are selected and removed from the population, the composition of the remaining population changes. This has the effect of altering the chances of different events or outcomes.
To illustrate this, let's consider an example. Let's say we have a bag with 5 red balls and 5 blue balls.
If we sample with replacement, after each draw, we put the selected ball back into the bag. So each time we draw, the number of balls and their proportion remains the same. Therefore, the probability of selecting a red or blue ball in each draw stays constant.
However, if we sample without replacement, after each draw, the selected ball is not returned. This means that the number of balls and their proportion changes after each draw. For example, if we select a red ball in the first draw, there are now only 4 red balls and 5 blue balls left in the bag. This changes the probabilities for subsequent draws.
In summary, sampling without replacement affects the probability of events by changing the composition of the population being sampled, whereas sampling with replacement keeps the probabilities constant throughout the sampling process.