distribution in the limiting case when the balls are drawn without replacement.
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When drawing mm balls with replacement, the probability that the maximum is equal to a value (kk) is: the probability that all balls are at most that value minus the probability that all balls are less than that value.
P(Y=k)=P(max{Xi}m≤k)−P(max{Xi}m≤k−1)P(Y=k)=P(max{Xi}m≤k)−P(max{Xi}m≤k−1)
When drawing with replacement, you need to find the probability of selecting a ball of that value and m−1m−1 balls of the k−1k−1 less than that value, out of all the ways to select any mm of nn balls.
When drawing mm balls with replacement, the probability that the maximum is equal to a value (kk) is: the probability that all balls are at most that value minus the probability that all balls are less than that value.
P(Y=k)=P(max{Xi}m≤k)−P(max{Xi}m≤k−1)P(Y=k)=P(max{Xi}m≤k)−P(max{Xi}m≤k−1)
When drawing with replacement, you need to find the probability of selecting a ball of that value and m−1m−1 balls of the k−1k−1 less than that value, out of all the ways to select any mm of nn balls.
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