Question

An unbiased dice with faces 1,2,3,4,5 and 6 is round 4 times. Out of four face values obtained the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 all the 4 times is
A) $\dfrac{16}{81}$
B) $\dfrac{1}{81}$
C) $\dfrac{80}{81}$
D) $\dfrac{65}{81}$

Answer 1

Hint: First find the probability of getting less than 2 or greater than 5. After that subtract the value from 1 to get the probability of not getting less than 2 or greater than 5. Then for the probability of not getting less than 2 and greater than 5 in 4 rolls, multiply the probability of not getting less than 2 or greater than 5 on one roll of the dice. Then solve it to get the desired result.

Complete step by step answer:
The total outcomes of the one dice are 6.
The probable outcome of getting less than 2 is,
$\Rightarrow n\left( X < 2 \right)=1$
The probable outcome of getting greater than 5 is,
$\Rightarrow n\left( X > 5 \right)=1$
Now, the probability of getting less than 2 or greater than 5 is,
$\Rightarrow P\left( X < 2,X > 6 \right)=\dfrac{2}{6}$
Cancel out the common factors,
$\Rightarrow P\left( X < 2,X > 6 \right)=\dfrac{1}{3}$
Then, the probability of not less than 2 or greater than 5 is,
$\Rightarrow P\left( 2\le X\le 5 \right)=1-\dfrac{1}{3}$
Take the LCM,
$\Rightarrow P\left( 2\le X\le 5 \right)=\dfrac{3-1}{3}$
On simplifications,
$\Rightarrow P\left( 2\le X\le 5 \right)=\dfrac{2}{3}$
Thus, for the 4 rolls, the probability will be,
$\Rightarrow P\left( 2\le X\le 5 \right)=\dfrac{2}{3}\times \dfrac{2}{3}\times \dfrac{2}{3}\times \dfrac{2}{3}$
Multiply the terms,
$\Rightarrow P\left( 2\le X\le 5 \right)=\dfrac{16}{81}$

Therefore, the required probability is $\dfrac{16}{81}$.Hence, option (A) is correct.

Note:
Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e. how likely they are to happen, using it.
The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favorable outcomes and the total number of outcomes.
$P\left( E \right)=\dfrac{n\left( A \right)}{n\left( S \right)}$
A probability of 0 means that an event is impossible.
A probability of 1 means that an event is certain.
An event with a higher probability is more likely to occur.
Probabilities are always between 0 and 1.