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 6. Determine whether E and F are independent events

Of course they are not independent. The ways to get six total on both die are

5,1
1,5
4,2
2,4
3,3

or five ways. If the first three is secluded, there are only four ways to get six.

Pr(E)=1/6
Pr(F)=5/36
Pr(F|E)= 1/6

Two events, E and F, are independent if the fact that E occurs does not affect the probability of F occurring. Here, The probability of F depends on whether or not E has occurred.

To determine if events E and F are independent, we need to check if the probability of their intersection is equal to the product of their probabilities.

First, let's find the probability of event E occurring. Since there are 6 equally likely outcomes for the first die, and only one of those outcomes is a 3, the probability of E is 1/6.

Next, let's find the probability of event F occurring. To calculate this, we need to find all the ways the sum of two dice can be 6. We have the following possibilities: (1,5), (2,4), (3,3), (4,2), (5,1). This means that out of the 36 possible outcomes, there are 5 outcomes that result in a sum of 6. Therefore, the probability of F is 5/36.

Now, let's find the probability of the intersection of E and F occurring. Since event E includes only one outcome (3) and event F includes two outcomes (3,3 and 4,2), the intersection of E and F contains only one outcome, which is (3,3). Therefore, the probability of the intersection is 1/36.

Finally, we check if the probability of the intersection is equal to the product of the probabilities of E and F.

(1/6) * (5/36) = 5/216

Since 1/36 is not equal to 5/216, we can conclude that events E and F are not independent events.

To determine whether events E and F are independent, we need to check if the occurrence of one event affects the probability of the other event occurring.

To calculate the probability of E, we need to determine the number of outcomes where the number on the first die is a 3, and divide it by the total number of possible outcomes.

There is only one outcome where the number on the first die is a 3: (3,1), (3,2), (3,3), (3,4), (3,5), (3,6). This is because the first die has six sides, and only one side has the number 3. The second die can land on any number from 1 to 6.

So, the probability of event E, denoted as P(E), is 1/6.

To calculate the probability of F, we need to determine the number of outcomes where the sum of the numbers on the two dice is 6, and divide it by the total number of possible outcomes.

The pairs that sum to 6 are: (1,5), (2,4), (3,3), (4,2), (5,1). There are five such pairs.

So, the probability of event F, denoted as P(F), is 5/36.

Now, to check if events E and F are independent, we need to see if the probability of both events occurring together, P(E and F), is equal to the product of their individual probabilities, P(E) * P(F).

The probability that both events occur, P(E and F), is the probability of rolling a (3,3), which is 1/36.

Calculating P(E) * P(F), we get (1/6) * (5/36) = 5/216.

Since P(E and F) is not equal to P(E) * P(F), we can conclude that events E and F are not independent.

In summary, the events E (the first die showing a 3) and F (the sum of the numbers is 6) are not independent events.