In probability, the symbol '0' is commonly used to represent an impossible event. An impossible event is an event that cannot occur under any circumstances.

The probability of an impossible event is always 0. This is because an impossible event cannot occur, so the probability of its occurrence is zero.

The probability of a certain event is always 1. This is because a certain event is guaranteed to occur, so the probability of its occurrence is 1.

In a probability distribution, the value of '0' often indicates the probability of an impossible event. This is because an impossible event has a probability of zero.

The expected value of a random variable that takes the value '0' with probability 'p' is 0. This is because the expected value is calculated by multiplying each possible value of the random variable by its probability and then summing the results. In this case, the only possible value is '0', and its probability is 'p', so the expected value is 0 * p = 0.

In a binomial distribution with parameters 'n' and 'p', the probability of obtaining exactly '0' successes is given by (1-p)^n. This is because the probability of success is 'p' and the probability of failure is 1-p. To obtain exactly '0' successes, we need to have 'n' consecutive failures, which has a probability of (1-p)^n.

The probability of obtaining at least one success in a sequence of 'n' independent Bernoulli trials with probability of success 'p' is given by 1 - (1-p)^n. This is because the probability of obtaining exactly '0' successes is (1-p)^n, and the probability of obtaining at least one success is the complement of this, which is 1 - (1-p)^n.

In a normal distribution, the probability of obtaining a value less than the mean 'μ' is 0.5. This is because the normal distribution is symmetric around the mean, so the probability of obtaining a value less than 'μ' is the same as the probability of obtaining a value greater than 'μ'.

In a normal distribution, the probability of obtaining a value greater than 'μ + σ' is approximately 0.1587. This can be calculated using the standard normal distribution (z-distribution) and the cumulative distribution function (CDF).

In a chi-square distribution with 'k' degrees of freedom, the probability of obtaining a value less than 'χ²_0.05,k' is 0.05. This is because the chi-square distribution is used for hypothesis testing, and the value 'χ²_0.05,k' is the critical value for a significance level of 0.05.

In an F-distribution with 'k1' and 'k2' degrees of freedom, the probability of obtaining a value greater than 'F_0.05,k1,k2' is 0.05. This is because the F-distribution is used for hypothesis testing, and the value 'F_0.05,k1,k2' is the critical value for a significance level of 0.05.

In a t-distribution with 'ν' degrees of freedom, the probability of obtaining a value less than 't_0.05,ν' is 0.05. This is because the t-distribution is used for hypothesis testing, and the value 't_0.05,ν' is the critical value for a significance level of 0.05.

In a non-central chi-square distribution with 'k' degrees of freedom and non-centrality parameter 'λ', the probability of obtaining a value less than 'χ²_0.05,k,λ' is 0.05. This is because the non-central chi-square distribution is used for hypothesis testing, and the value 'χ²_0.05,k,λ' is the critical value for a significance level of 0.05.

In a non-central t-distribution with 'ν' degrees of freedom and non-centrality parameter 'δ', the probability of obtaining a value less than 't_0.05,ν,δ' is 0.05. This is because the non-central t-distribution is used for hypothesis testing, and the value 't_0.05,ν,δ' is the critical value for a significance level of 0.05.