A pair of fair dice is rolled. Let E denote the event that the number landing uppermost on the first die is a 3, and let F denote the event that the sum of the numbers landing uppermost is 7. Determine whether E and F are independent events.
For E and F to be independent, doesn't P(E l E) have to be equal to P(F)?
I know for F, there are 6 ways to sum up 7 (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).
Sorry, I meant doesn't P (F l E) have to be equal to P (F)? Or vice versa ( P(E l F) be equal to P(E)?
If the first die is 3, then to sum as 7, the second die has to be 4. Is that independence?
To determine whether events E (landing uppermost on the first die is a 3) and F (sum of the numbers landing uppermost is 7) are independent, we need to calculate the probabilities of these events and compare them.
First, let's calculate P(E), the probability of event E occurring.
Since a pair of fair dice is rolled, the number on each die has a probability of 1/6 of landing uppermost. Therefore, the probability of the first die landing on 3 is 1/6.
Next, let's calculate P(F), the probability of event F occurring.
As you correctly mentioned, there are 6 possible ways to obtain a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).
Since there are a total of 36 possible outcomes when two dice are rolled (6 options for the first die multiplied by 6 options for the second die), the probability of getting a sum of 7 is 6/36, which simplifies to 1/6.
Now we can check if these events are independent by comparing P(E l F) with P(E) and P(F).
To find P(E l F), the probability of events E and F both occurring, we need to consider how many outcomes satisfy both conditions. Only one of the six outcomes for event F (specifically, (3,4)) satisfies event E as well, since it also has a 3 on the first die. Therefore, the probability of E and F occurring together is 1/36.
Since P(E l F) is not equal to P(E) multiplied by P(F), these events are not independent. In this case, knowing that the sum of the numbers is 7 affects the probability of the first die landing on 3.