Sampling With Replacement

When we randomly select an element, needed for our study, from a population of interest and mix it again with the same population so that it may be selected again. We say that we did sampling with replacement. In sampling with replacement the same element may be selected more than once from the population of interest. In this method of sampling all the possible events are independent of each other and their covariance is zero. In other words the occurrence of one event has no effect upon the occurrence of other event. And the probabilities of their occurrence remains equal regardless the frequency of each event in the process. For example in throwing a dice, each number (1-6) on the dice has 1/6 probability of being selected, no matter how many times we throw it. For sampling without replacement click here.

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Sampling Without Replacement

If an element cannot be selected again after being selected once, we say that we did sampling without replacement. For example when we pick out a ball from a jar containing five balls of different colors and put it aside instead of placing it back in the same jar. Then pick out the second ball from the jar in the same way and continue this process until the last ball, then the probability of the first ball will be 1/5; the probability of the 2nd ball will be 1/4, the probability of the 3^rd ball will be 1/3, the probability of the 4^th  ball will be 1/2 while the probability of 5^th ball will be 1.

We see here that the probability of every next ball varies as we pick out more and more balls from the jar. In other words, as the total number of balls in the jar decreases their probability of being selected increase. Here each event, picking a ball from the jar, has an effect upon subsequent events. So we can say that they are not independent and mutually exclusive of each other and their covariance is not equal to zero. For sampling with replacement click here.

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