What is the sum of the probabilities in a probability distribution?

Question

maul124uk · Accepted Answer

The sum of the probabilities in a probability distribution is always 1.
A probability distribution is a collection of probabilities that defines the likelihood of observing all of the various outcomes of an event or experiment. Based on this definition, a probability distribution has two important properties that are always true:
Each probability in the distribution must be of a value between 0 and 1.
The sum of all the probabilities in the distribution must be equal to 1.
An example: You could define a probability distribution for the observation for the number displayed by a single roll of a die. The probability that the die with show a 1 is

\frac{1}{6}

.
Thats

SlabydouluS62 · Accepted Answer

The sum of the probabilities in a probability distribution is always 1. A probability distribution is a collection of probabilities that defines the likelihood of observing all of the various outcomes of an event or experiment.

RizerMix · Accepted Answer

Answer: 1Explanation:In probability theory, a probability distribution is a function that describes the likelihood of obtaining possible outcomes of a random variable. The sum of the probabilities in a probability distribution is always equal to 1.Formally, if X is a random variable with a probability distribution function p(x), defined over a set of possible values 𝒳, then we have:∑x∈𝒳p(x)=1This equation is known as the probability normalization condition, which states that the total probability of all possible outcomes of X is equal to 1. This condition is a consequence of the axioms of probability theory and is a fundamental property of probability distributions.To illustrate this, consider a simple example of a fair six-sided die. The probability of each possible outcome is 1/6, since there are six equally likely outcomes. Thus, the sum of the probabilities in the probability distribution is:∑i=1616=1which confirms that the total probability of all possible outcomes is indeed equal to 1.

Jeffrey Jordon · Accepted Answer

The sum of probabilities in a probability distribution is always equal to

1

is by considering the concept of a sample space. The sample space

𝒮

is the set of all possible outcomes of an experiment or random process.
For example, if we toss a fair coin, the sample space is

𝒮 = {H, T}

, where

H

represents heads and

T

represents tails. The probability distribution function assigns a probability to each element in the sample space, such that

\sum_{s \in 𝒮} p (s) = 1

, where

p (s)

is the probability of outcome

s

.
Intuitively, the sum of probabilities must be equal to

1

because the probability of the sample space is

1

. This means that one of the possible outcomes in the sample space must occur, and the sum of the probabilities of all possible outcomes must add up to

1

.
In other words, the probability distribution function must assign probabilities that cover the entire sample space and leave no room for any other outcomes. If the sum of probabilities was less than

1

, it would mean that some possible outcomes are not accounted for, and if the sum of probabilities was greater than

1

, it would mean that some outcomes have been double-counted.
Therefore, the sum of probabilities in a probability distribution must always be equal to

1

, by definition.

Vasquez · Accepted Answer

In probability theory, a probability distribution is a function that describes the likelihood of obtaining each possible value of a random variable. The sum of the probabilities in a probability distribution must be equal to 1. We can express this mathematically as:

\sum_{i} P (X = i) = 1

where

P (X = i)

represents the probability of the random variable

X

taking the value

i

, and the sum is taken over all possible values of

X

.
To see why this is true, consider the following. If we add up the probabilities of all possible outcomes of an experiment, we should get a total probability of 1, since one of those outcomes must occur. This is the basic principle of probability.
For example, suppose we have a coin that is equally likely to land heads or tails, and we define the random variable

X

to be the number of heads obtained in two tosses of the coin. Then the possible values of

X

are 0, 1, and 2, and the corresponding probabilities are:

P (X = 0) = \frac{1}{4}

P (X = 1) = \frac{1}{2}

P (X = 2) = \frac{1}{4}

The sum of these probabilities is:

\sum_{i = 0}^{2} P (X = i) = P (X = 0) + P (X = 1) + P (X = 2) = \frac{1}{4} + \frac{1}{2} + \frac{1}{4} = 1

which confirms that the sum of the probabilities in a probability distribution must be equal to 1.

What is the sum of the probabilities in a probability

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