We are given five outcomes $w_1, w_2, w_3, w_4, w_5$, where:

• $w_2 = 0.1$
• $w_3 = 0.1$
• $w_4 = 0.25$
• $w_1 = w_5$

Since the total probability must sum to 1, we can write:

$w_1 + w_2 + w_3 + w_4 + w_5 = 1$

Substitute the known values and the fact that $w_1 = w_5$:

$w_1 + 0.1 + 0.1 + 0.25 + w_1 = 1$

Simplifying this equation:

$2w_1 + 0.45 = 1$

Subtract 0.45 from both sides:

$2w_1 = 0.55$

Now divide by 2:

$w_1 = 0.275$

Therefore, $w_1 = w_5 = 0.275$.

Would you like further clarification on this?

Here are 5 related questions you might find interesting:

1. What if the total probability did not sum to 1?
2. How do we interpret the sum of probabilities in other statistical distributions?
3. Can you extend this reasoning to scenarios with more than five outcomes?
4. What if $w_1$ and $w_5$ were not equal? How would this change the outcome?
5. How can we apply this concept to continuous probability distributions?

Tip: Always ensure that the sum of probabilities in any discrete distribution adds up to 1!