Question

A dice is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 appeared at least once?

Answer 1

Hint: In this particular type of questions use the concept that the conditional probability which is asked is the ratio of the probability of the intersection of events (i.e. the sum of the digits is 6 and digit 4 appeared at least once) so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given:
A dice is thrown twice and the sum of the numbers appearing is observed to be 6.
Now we have to find the conditional probability that the number 4 appeared at least once.
Let, A be the event that the sum of the numbers is 6.
So the number of ways that the sum of the dice is 6 = (1, 5), (2, 4), (3, 3), (4, 2), (5, 1).
Therefore, A = (1, 5), (2, 4), (3, 3), (4, 2), (5, 1).
So as we see that there are 5 possible ways so that the sum of the numbers is 6.
As we know that when two dice are thrown the possible number of cases are ($6 \times 6$) = 36.
So the probability to get the sum, P (A) = $\dfrac{{{\text{favorable number of outcomes}}}}{{{\text{total number of outcomes}}}}$
$ \Rightarrow P\left( A \right) = \dfrac{5}{{36}}$
Now let (E) be the event such that 4 appears once.
So the possible cases when two dice are throw one by one,
E = (1, 4), (2, 4), (3, 4), (4, 4), (5, 4), (6, 4), (4, 1), (4, 2), (4, 3), (4, 5), (4, 6).
So there are 11 possible cases such that 4 appear once.
Now the intersection of event A and E, i.e. $A \cap E$= (2, 4) and (4, 2)
So the probability of, $P\left( {A \cap E} \right) = \dfrac{2}{{36}}$
Now the conditional probability P (E/A) such that the sum of the numbers is 6 and digit 4 appeared once is,
$ \Rightarrow P\left( {\dfrac{E}{A}} \right) = \dfrac{{P\left( {E \cap A} \right)}}{{P\left( A \right)}}$
Now substitute the values we have,
$ \Rightarrow P\left( {\dfrac{E}{A}} \right) = \dfrac{{\dfrac{2}{{36}}}}{{\dfrac{5}{{36}}}} = \dfrac{2}{5}$
So this is the required probability.

Note: Whenever we face such types of questions the key concept we have to remember is that the probability is the ratio of favorable number of outcomes to the total number of outcomes, so first find out the favorable number of outcomes then total number of outcomes then divide them to calculate the probabilities of the events as above.