Suppose you have a bag with 11 balls in it, and you have placed 11 buckets on the floor (where each bucket has a label affixed to it; say, the buckets are labeled 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11). Now, consider the following experiment. You draw a ball from the bag, toss the ball into one of the buckets, and record the bucket in which the ball landed (for simplicity, assume that the ball has to land in a bucket; i.e., you cannot “miss” all of the buckets and hit the floor, or anything like that). You repeat this procedure until your bag of balls is empty (at which point the experiment has concluded). Let S denote the sample space for this experiment, and define the events E and F as follows. E: All of the balls are in one bucket at the conclusion of the experiment. F: No bucket is empty at the conclusion of the experiment. Write down the set E. (This may be a bit tedious—although certainly reasonable—to do explicitly; it can, however, be done concisely—and precisely—using set-builder notation.)
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
Suppose you have a bag with 11 balls in it, and you have placed 11 buckets on the
floor (where each bucket has a label affixed to it; say, the buckets are labeled 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11). Now, consider the following experiment. You draw a ball from the bag, toss the ball into one of the
buckets, and record the bucket in which the ball landed (for simplicity, assume that the ball has to land
in a bucket; i.e., you cannot “miss” all of the buckets and hit the floor, or anything like that). You repeat
this procedure until your bag of balls is empty (at which point the experiment has concluded).
Let S denote the
E: All of the balls are in one bucket at the conclusion of the experiment.
F: No bucket is empty at the conclusion of the experiment.
Write down the set E. (This may be a bit tedious—although certainly reasonable—to do explicitly;
it can, however, be done concisely—and precisely—using set-builder notation.)
Introduction:
It may be assumed that each ball tossed by the experimenter lands with equal probability into any one of the 11 buckets. Further, all the balls must fall into one of the buckets.
The first ball can land in any of the 11 buckets.
The second ball can land in any of the 11 buckets, independent of where the first ball had landed.
The third ball can land in any of the 11 buckets, independent of where the second ball had landed.
Proceeding in this manner, it can be said that the eleventh ball can land in any of the 11 buckets, independent of where the previous 10 balls had landed.
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