Let's first analyze the table provided in the image:

We are missing the relative frequency for outcome 5, but since the sum of relative frequencies must equal 1, we can calculate it.

Step 1: Complete the table

To find the missing frequency: $\text{Sum of given frequencies} = 0.2 + 0.2 + 0.1 + 0.1 = 0.6$ Thus, the missing relative frequency for outcome 5 is: $\text{Relative frequency of outcome 5} = 1 - 0.6 = 0.4$

Now the complete table is:

Step 2: Calculate the probabilities

(a) $P(\{1, 3, 5\})$

The probability of this event is the sum of the relative frequencies for outcomes 1, 3, and 5: $P(\{1, 3, 5\}) = 0.2 + 0.1 + 0.4 = 0.7$

(b) $P(E')$, where $E = \{1, 2, 3\}$

First, the probability of $E$ is the sum of the relative frequencies for outcomes 1, 2, and 3: $P(E) = 0.2 + 0.2 + 0.1 = 0.5$ The complement $E'$ represents the outcomes not in $E$, which are 4 and 5: $P(E') = P(\{4, 5\}) = 0.1 + 0.4 = 0.5$

Final Answers:

• (a) $P(\{1, 3, 5\}) = 0.7$
• (b) $P(E') = 0.5$

Would you like more details on any part of this solution?

Here are some follow-up questions you can explore:

1. How do you compute the complement of a probability event?
2. How do relative frequencies relate to empirical probabilities?
3. What does it mean for the sum of probabilities to equal 1?
4. How does one calculate probabilities for compound events?
5. Can the sum of relative frequencies exceed 1 in any case?

Tip: When calculating probabilities, always make sure that all events considered are mutually exclusive, and their total probability sums to 1.