Which of the following is true about a probability distribution?
the sum of all possible outcomes must equal 1.
the outcomes must be mutually exclusive.
the probability of each outcome must be between 0 and 1 inclusive.
all of the above.
The correct answer and explanation is :
The correct answer is “all of the above.”
A probability distribution describes the likelihood of various outcomes in a probabilistic event or experiment. Let’s break down the three components to explain why all of them are true:
- The sum of all possible outcomes must equal 1:
This is a fundamental characteristic of probability distributions. For any experiment or random variable, the total probability of all possible outcomes must always sum to 1. This reflects the certainty that one of the possible outcomes will indeed occur. If the sum of all probabilities is less than 1, it indicates that there are some outcomes missing or unaccounted for. If the sum exceeds 1, it means there is overcounting of probabilities. This rule ensures the probability distribution is valid. - The outcomes must be mutually exclusive:
In the context of a probability distribution, mutually exclusive outcomes are those that cannot occur at the same time. For example, in a dice roll, the outcome cannot be both 1 and 2 simultaneously. In probability distributions, the events must not overlap; if one outcome occurs, the others cannot. This exclusivity is essential because the total probability of all possible outcomes is calculated assuming they are separate, distinct events. - The probability of each outcome must be between 0 and 1 inclusive:
The probability of each individual outcome in a probability distribution must be between 0 and 1. This is because probability is a measure of the likelihood of an event happening, and it cannot be negative (probability cannot be less than 0) or exceed 1 (a probability greater than 1 would indicate certainty beyond possible outcomes). A probability of 0 means an event will not occur, while a probability of 1 means the event is certain to occur. Any probability in between represents varying degrees of likelihood.
Thus, all three components listed are essential characteristics of a valid probability distribution, making “all of the above” the correct answer.