Question

g A fair coin is tossed 20 times. The number of heads observed is the count X of successes. Give the distribution of X . Choose the correct answer; (X is the random variable and is the Binomial distribution):

Answer 1

We assume that we have a fair coin that is
`p(Head)=p(Tails)=0.5`

For this case we define the random variable X as "number of heads observed in 20 times". The distribution for X is given by:

`X \sim Binom(n=20, p=0.5)`

The probability mass function for the Binomial distribution is given as:

`P(X)=(nCx)(p)^x (1-p)^(n-x)`

Where (nCx) means combinatory and it's given by this formula:

`nCx=(n!)/((n-x)! x!)`

Explanation:

Previous concepts

A Bernoulli trial is "a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted". And this experiment is a particular case of the binomial experiment.

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

The probability mass function for the Binomial distribution is given as:

`P(X)=(nCx)(p)^x (1-p)^(n-x)`

Where (nCx) means combinatory and it's given by this formula:

`nCx=(n!)/((n-x)! x!)`

Solution to the problem

We assume that we have a fair coin that is
`p(Head)=p(Tails)=0.5`

For this case we define the random variable X as "number of heads observed in 20 times". The distribution for X is given by:

`X \sim Binom(n=20, p=0.5)`

The probability mass function for the Binomial distribution is given as:

`P(X)=(nCx)(p)^x (1-p)^(n-x)`

Where (nCx) means combinatory and it's given by this formula:

`nCx=(n!)/((n-x)! x!)`