Question

Suppose that a die (with faces marked 1 to 6) is loaded in such a manner that for \[k = 1,2,3,...,6\] the probability of the face marked \[k\] turning up when the die is tossed is proportional to \[k\]. The probability of the event that the outcome of a toss of the die will be an even number is equal to
A. \[\dfrac{1}{2}\]
B. \[\dfrac{4}{7}\]
C. \[\dfrac{2}{5}\]
D. \[\dfrac{1}{21}\]

Answer 1

Hint: In this question, we need to find the probability of the event that the outcome of a toss of the die will be an even number. For this, we need to find the total cases and the favorable outcomes. For finding the probability, we have to take the ratio of favorable outcomes to the total outcomes.

Formula used: The following formula of probability is used to solve the given question.
Probability = $\dfrac{\text{Favourable outcomes}}{\text{Total outcomes}}$

Complete step-by-step solution:
We know that a die (with faces marked 1 to 6) is loaded in such a manner that for \[k = 1,2,3,...,6\] the probability of the face marked \[k\] turning up when the die is tossed is proportional to \[k\].
Let us find the total outcomes.
As die has 6 faces marked 1 to 6, the total number of outcomes is 6 such as 1, 2, 3, 4, 5, and 6.
Now, we will find the favorable outcomes for the event that the outcome of a toss of the die will be an even number.
So, the favorable outcomes are 2, 4, and 6.
In simple terms, we can say that
Total outcomes \[ = \{ 1,2,3,4,5,6\} \]
Favourable outcomes \[ = \{ 2,4,6\} \]
That means the total number of favorable outcomes is 3.
Thus, the probability is given by
Probability = $\dfrac{\text{Favourable outcomes}}{\text{Total outcomes}}$
Thus, we get
\[P = \dfrac{3}{6}\]
By simplifying, we get
\[P = \dfrac{1}{2}\]
Hence, the probability of the event that the outcome of a toss of the die will be an even number is \[\dfrac{1}{2}\].

Therefore, the correct option is (A).

Note: Many students generally make mistakes in finding favorable outcomes. They may take the reverse ratio to find the probability such as the ratio of total outcomes to favorable outcomes to get probability but this gives the wrong result.