Let�� �S={s1,s2,s3,s4}

�be a sample space with probability distribution P,�
given by� �
P(s1)= 0.5, P(s2)= 0.25, P(s3)= 0.125, P(s4)= 0.125.

There are sixteen possible events that can be formed from the elements of� 'S'.
Compute the probability and surprise of these events.

I assume that each of the sixteen events has two outcomes of S, namely

P(s1,s1), P(s1,s2).....P(s4,s4).
By the multiplication rule, and assuming that the two outcomes are independent, we have
P(s1,s1)=P(s1)P(s1)=0.5*0.5;=0.25
....
P(s4,s4)=P(s4)P(s4)=0.125*0.125=0.015625
Note: The sum of probabilities of the 16 events should add up to 1.
Not sure about the surprise part.

To compute the probability of each event, we need to sum up the probabilities of the individual outcomes that make up the event.

Let's go through each event and calculate its probability:

1. Event: {} (Empty event)
Probability: P({}) = 0

2. Event: {s1}
Probability: P({s1}) = P(s1) = 0.5

3. Event: {s2}
Probability: P({s2}) = P(s2) = 0.25

4. Event: {s3}
Probability: P({s3}) = P(s3) = 0.125

5. Event: {s4}
Probability: P({s4}) = P(s4) = 0.125

6. Event: {s1, s2}
Probability: P({s1, s2}) = P(s1) + P(s2) = 0.5 + 0.25 = 0.75

7. Event: {s1, s3}
Probability: P({s1, s3}) = P(s1) + P(s3) = 0.5 + 0.125 = 0.625

8. Event: {s1, s4}
Probability: P({s1, s4}) = P(s1) + P(s4) = 0.5 + 0.125 = 0.625

9. Event: {s2, s3}
Probability: P({s2, s3}) = P(s2) + P(s3) = 0.25 + 0.125 = 0.375

10. Event: {s2, s4}
Probability: P({s2, s4}) = P(s2) + P(s4) = 0.25 + 0.125 = 0.375

11. Event: {s3, s4}
Probability: P({s3, s4}) = P(s3) + P(s4) = 0.125 + 0.125 = 0.25

12. Event: {s1, s2, s3}
Probability: P({s1, s2, s3}) = P(s1) + P(s2) + P(s3) = 0.5 + 0.25 + 0.125 = 0.875

13. Event: {s1, s2, s4}
Probability: P({s1, s2, s4}) = P(s1) + P(s2) + P(s4) = 0.5 + 0.25 + 0.125 = 0.875

14. Event: {s1, s3, s4}
Probability: P({s1, s3, s4}) = P(s1) + P(s3) + P(s4) = 0.5 + 0.125 + 0.125 = 0.75

15. Event: {s2, s3, s4}
Probability: P({s2, s3, s4}) = P(s2) + P(s3) + P(s4) = 0.25 + 0.125 + 0.125 = 0.5

16. Event: {s1, s2, s3, s4}
Probability: P({s1, s2, s3, s4}) = P(s1) + P(s2) + P(s3) + P(s4) = 0.5 + 0.25 + 0.125 + 0.125 = 1

To compute the surprise of each event, we can use the formula: surprise = -log2(probability)

Let's calculate the surprise for each event using the probabilities we computed above:

1. Event: {}
Probability: 0
Surprise: -log2(0) = undefined

2. Event: {s1}
Probability: 0.5
Surprise: -log2(0.5) = 1

3. Event: {s2}
Probability: 0.25
Surprise: -log2(0.25) = 2

4. Event: {s3}
Probability: 0.125
Surprise: -log2(0.125) = 3

5. Event: {s4}
Probability: 0.125
Surprise: -log2(0.125) = 3

6. Event: {s1, s2}
Probability: 0.75
Surprise: -log2(0.75) ≈ 0.415

7. Event: {s1, s3}
Probability: 0.625
Surprise: -log2(0.625) ≈ 0.677

8. Event: {s1, s4}
Probability: 0.625
Surprise: -log2(0.625) ≈ 0.677

9. Event: {s2, s3}
Probability: 0.375
Surprise: -log2(0.375) ≈ 1.415

10. Event: {s2, s4}
Probability: 0.375
Surprise: -log2(0.375) ≈ 1.415

11. Event: {s3, s4}
Probability: 0.25
Surprise: -log2(0.25) = 2

12. Event: {s1, s2, s3}
Probability: 0.875
Surprise: -log2(0.875) ≈ 0.184

13. Event: {s1, s2, s4}
Probability: 0.875
Surprise: -log2(0.875) ≈ 0.184

14. Event: {s1, s3, s4}
Probability: 0.75
Surprise: -log2(0.75) ≈ 0.415

15. Event: {s2, s3, s4}
Probability: 0.5
Surprise: -log2(0.5) = 1

16. Event: {s1, s2, s3, s4}
Probability: 1
Surprise: -log2(1) = 0

This completes the calculation of the probabilities and surprises of the sixteen possible events formed from the elements of the sample space S.

To compute the probability and surprise of each event, we will first need to list down all the possible events that can be formed from the elements of the sample space S.

Since S has four elements, there are 2^4 = 16 possible events. These events can be represented using subsets of S. Here are all the possible events:

Event 1: Empty set (∅)
Event 2: {s1}
Event 3: {s2}
Event 4: {s3}
Event 5: {s4}
Event 6: {s1, s2}
Event 7: {s1, s3}
Event 8: {s1, s4}
Event 9: {s2, s3}
Event 10: {s2, s4}
Event 11: {s3, s4}
Event 12: {s1, s2, s3}
Event 13: {s1, s2, s4}
Event 14: {s1, s3, s4}
Event 15: {s2, s3, s4}
Event 16: {s1, s2, s3, s4}

Now, let's compute the probability and surprise of each of these events.

1. Event 1: Empty set (∅)
Probability: P(∅) = 0 (as there are no elements in ∅)
Surprise: -log2(P(∅)) = -log2(0) (undefined)

2. Event 2: {s1}
Probability: P({s1}) = P(s1) = 0.5
Surprise: -log2(P({s1})) = -log2(0.5) = 1

3. Event 3: {s2}
Probability: P({s2}) = P(s2) = 0.25
Surprise: -log2(P({s2})) = -log2(0.25) = 2

4. Event 4: {s3}
Probability: P({s3}) = P(s3) = 0.125
Surprise: -log2(P({s3})) = -log2(0.125) = 3

... and so on for the remaining events.

By computing the probability and surprise for each event, you can determine their respective values.