Question

A coin is tossed and a dice is rolled. The probability that the coin shows the head and the dice shown is $6$is $?$
1)$$$$\dfrac{1}{8}
2)$$$$\dfrac{1}{12}
3)$$$$\dfrac{1}{2}
4)$$$$1

Answer 1

Hint : In this type of question you can first find the probability of the first event occurring and then find the probability of the second event, then you should try to observe that the multiple events occurring are dependent or independent and solve accordingly. Here we have independent events .

Complete step-by-step solution:
Let $A$ be the event of tossing a coin.
Now we need to find the probability that the coin shows the head, let say it by $P(A)$.
Therefore, we will apply basic formula to find the probability of the favourable outcome out of total outcomes and that is given below:
$Probability=\dfrac{Number\\_Of\\_Favourable\\_Outcomes}{Total\\_Number\\_Of\\_Outcomes}$
And to find the probability of Event $A$we have,
Number of heads in a coin$=1$
Total number of faces in coin$=2$
Therefore, Probability of getting Head after tossing a coin will be:
$P(A)=\dfrac{1}{2}$
Similarly,
Let $B$ be the event of rolling a dice.
Now we need to find the probability that a dice shows $6$, let say it by $P(B)$.
Therefore, we will again apply basic formula to find the probability of the favourable outcome out of total outcomes and that is given below:
$Probability=\dfrac{Number\\_Of\\_Favourable\\_Outcomes}{Total\\_Number\\_Of\\_Outcomes}$
And to find the probability of Event $B$we have,
Number of faces having $6$ in a dice$=1$
Total number of faces in a dice$=6$
Therefore, Probability of getting $6$ after rolling a dice will be:
$P(B)=\dfrac{1}{6}$
Till now we found the probability of the first event occurring and then found the probability of the second event occurring, now if we observe that the multiple events occurring here are independent of each other, that means the probability of either of the events cannot be affected by the other one.
So we will simply Multiply probabilities of both the events to get our required answer as both the events occurring at the same time are independent of each other.
So, the required probability is,
$Probability=P(A)\times P(B)$
$=\dfrac{1}{2}\times \dfrac{1}{6}$
$=\dfrac{1}{12}$
So, option $(2)$ is correct.
Hence the final answer we got is option $(2)$.

Note: Two events can be dependent also if the outcome of the first event affects the outcome of the second event, so the probability gets changed. In that case one must find the probability using Multiplication theorem or Conditional Probability and various other concepts and logic.