Question

A die is thrown 20 times. Getting a number greater than 4 is considered a success. Find the mean and variance of the number of successes.

Answer 1

Hint: The binomial distribution formula helps to check the probability of getting “x” successes in “n” independent trials of a binomial experiment. The binomial distribution is a type of probability distribution in statistics that has two possible outcomes i.e. success and failure. The probability of failure is “1-x”. This distribution is also known as Bernoulli distribution. The performance of a fixed number of trails with a fixed probability of success on each trial is known as a Bernoulli trial.
Mean is the average of the given set of numbers.
Mean of binomial distribution is given by \[E\left( X \right){\text{ }} = {\text{ }}np\]
Variance is a measure of spread for distribution of a random variable that determines the degree to which the values of a random variable differ from the expected value.
Variance of binomial distribution is given by $Var(X) = np(1 - p)$

Complete step-by-step answer:
Clearly, the distribution of the ‘number of successes’ is a binomial distribution with n = 20
When we throw a dice number greater than 4 are (5,6)
So, the favourable outcomes = 2
And total outcomes = 6
Let the probability of getting a number greater than 4 = $p = \dfrac{2}{6} = \dfrac{1}{3}$
Therefore, probability of not getting number greater than 4 = q
$\begin{gathered}
  q = 1 - p \\
  q = 1 - \dfrac{1}{3} \\
  q = \dfrac{2}{3} \\
\end{gathered} $
Now, Mean = np and Variance = npq
Mean = $20 \times \dfrac{1}{3} = 6.66$
Variance = $20 \times \dfrac{1}{3} \times \dfrac{2}{3} = 4.44$
Hence, the mean and variance of the number of successes are 6.66 and 4.44 respectively.

Note: Conditions of binomial probability distribution:
The number of observations is fixed.
Each observation is independent.
Each observation represents one of two outcomes (“success or failure”)
The probability of “success” p is the same for each outcome.