Consider a dice with the property that the probability of a face with n dots showing up is proportional to n. The probability of the face with three dots showing up is_________.Correct answer is '0.10'. Can you explain this answer?

Question

Answer

Probability distribution of the dice

The given dice has the property that the probability of a face with n dots showing up is proportional to n. This means that the probabilities of getting different numbers of dots on the dice are not equally likely. Let's denote the probability of getting n dots as P(n).

Proportional probability

Since the probability is proportional to the number of dots, we can write the proportionality as:

P(n) ∝ n

This means that the probability of getting n dots is equal to some constant k multiplied by n:

P(n) = kn

To find the value of k, we can use the fact that the sum of all probabilities must be equal to 1. Therefore, we have:

∑ P(n) = 1

Substituting the expression for P(n), we get:

∑ kn = 1

Simplifying the equation, we have:

k ∑ n = 1

Since the dice has six faces with numbers from 1 to 6, we can write the sum as:

1k + 2k + 3k + 4k + 5k + 6k = 1

Calculating the value of k

Simplifying the equation further, we have:

21k = 1

Dividing both sides by 21, we get:

k = 1/21

Calculating the probability of getting three dots

Now that we know the value of k, we can calculate the probability of getting three dots. Substituting n=3 and k=1/21 into the expression for P(n), we have:

P(3) = (1/21) * 3

Simplifying the equation, we get:

P(3) = 3/21

Simplifying the fraction, we have:

P(3) = 1/7

Converting the fraction to decimal form, we get:

P(3) = 0.142857

Rounding the decimal to two decimal places, we have:

P(3) ≈ 0.14

Therefore, the probability of the face with three dots showing up is approximately 0.14.

Correct answer

The correct answer given is '0.10'. However, based on the calculation, the probability is approximately 0.14. Therefore, there seems to be an error in the answer provided.

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