Question

A die is thrown $100$ times, getting an even number is considered a success. The variance of the number of successes is
A.$10$
B.$25$
C.$18$
D.$11$

Answer 1

Hint : Probability is a branch of mathematics that deals with calculating the likelihood of a given event's occurrence, which is expressed as a number between $0$ and $1$ . An event with a probability of $1$ can be considered a sure event, an event with a probability of $5$ can be considered to have equal odds of occurring or not occurring and an event with a probability of $0$ can be considered an impossible event.

Complete step-by-step answer :
Sample Space: The sample pace associated with a random experiment is the set of all possible outcomes. An event is a subset of the sample space.
Event: An event E is said to occur on a particular trial of the experiment if the outcome observed is an element of the set E.
The probability formula is defined as the probability of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.
Probability (event) $= \dfrac{{Number\; of \;Favourable \;Outcomes}}{{Total \;number \; of \;outcomes}}$
Variance: The variance is a measure of variability. Variance tells you the degree of spread in your data set.
In throw of a dice, Sample space = \left\\{ {1,2,3,4,5,6} \right\\}
Probability of getting an even number $= \dfrac{3}{6} = \dfrac{1}{2}$
Therefore $p = \dfrac{1}{2}$
Hence $q = 1 - p = 1 - \dfrac{1}{2} = \dfrac{1}{2}$
Therefore the variance $= npq$
$= \left( {100} \right)\left( {\dfrac{1}{2}} \right)\left( {\dfrac{1}{2}} \right)$
$= 25$

So, the correct answer is “Option B”.

Note : The meaning of probability is basically the extent to which something is likely to happen. Remember about random experiment, sample space and favourable outcomes related to the given event. Probability of any event can be between $0$ and $1$ only. Probability of any event can never be greater than $1$ . Probability of any event can never be negative.