Question

What are the probabilities of multiple outcomes with three outcomes A, B, and C, where the probability of C is twice the probability of A, which is equal to the probability of B?

Answer 1

Let's define the probabilities of the three outcomes as follows:

- Let \( P(A) = p \)
- Let \( P(B) = p \)
- Let \( P(C) = 2p \)

Since the sum of the probabilities of all possible outcomes must equal 1, we can write the equation:

\[
P(A) + P(B) + P(C) = 1
\]

Substituting the defined probabilities into this equation gives:

\[
p + p + 2p = 1
\]

This simplifies to:

\[
4p = 1
\]

To find \( p \), we can solve for it:

\[
p = \frac{1}{4}
\]

Now we can determine the probabilities for each outcome:

- \( P(A) = p = \frac{1}{4} \)
- \( P(B) = p = \frac{1}{4} \)
- \( P(C) = 2p = 2 \cdot \frac{1}{4} = \frac{1}{2} \)

Thus, the probabilities of the outcomes A, B, and C are:

- \( P(A) = \frac{1}{4} \)
- \( P(B) = \frac{1}{4} \)
- \( P(C) = \frac{1}{2} \)