To determine if two events are independent, we need to check if the probability of their joint occurrence equals the product of their individual probabilities. The formula for independence is:
[tex]\[ P(A \text{ and } B) = P(A) \times P(B) \][/tex]
Let's use the given table and data to find out if any events are independent.
First, we'll calculate the probabilities for events A and Y, and E and Y.
### Step 1: Calculate Joint Probabilities
We need to find the probabilities for the joint occurrences of the events.
- [tex]\( P(A \text{ and } Y) \)[/tex] corresponds to the value in the (A, Y) cell:
[tex]\[ P(A \text{ and } Y) = \frac{5}{100} = 0.05 \][/tex]
- [tex]\( P(E \text{ and } Y) \)[/tex] corresponds to the value in the (E, Y) cell:
[tex]\[ P(E \text{ and } Y) = \frac{8}{100} = 0.08 \][/tex]
### Step 2: Calculate Marginal Probabilities
Next, we need to calculate the marginal probabilities for A, Y, and E.
- [tex]\( P(A) \)[/tex] is the total for row A divided by the grand total:
[tex]\[ P(A) = \frac{30}{100} = 0.30 \][/tex]
- [tex]\( P(Y) \)[/tex] is the total for column Y divided by the grand total:
[tex]\[ P(Y) = \frac{28}{100} = 0.28 \][/tex]
- [tex]\( P(E) \)[/tex] is the total for row E divided by the grand total:
[tex]\[ P(E) = \frac{20}{100} = 0.20 \][/tex]
### Step 3: Check Independence
Now we check for independence by comparing [tex]\( P(A \text{ and } Y) \)[/tex] and [tex]\( P(A) \times P(Y) \)[/tex]:
[tex]\[ P(A \text{ and } Y) = 0.05 \][/tex]
[tex]\[ P(A) \times P(Y) = 0.30 \times 0.28 = 0.084 \][/tex]
Since [tex]\( 0.05 \neq 0.084 \)[/tex], A and Y are not independent.
Similarly, we compare [tex]\( P(E \text{ and } Y) \)[/tex] and [tex]\( P(E) \times P(Y) \)[/tex]:
[tex]\[ P(E \text{ and } Y) = 0.08 \][/tex]
[tex]\[ P(E) \times P(Y) = 0.20 \times 0.28 = 0.056 \][/tex]
Since [tex]\( 0.08 \neq 0.056 \)[/tex], E and Y are not independent.
### Conclusion
Because neither A and Y nor E and Y are independent, and no other pairs of events are given in the question, we conclude that none of the events provided are independent. Hence, there are no pairs of independent events in the data given.
