There are 36 total outcomes for this experiment and there are six ways to roll a 7 with two dice: 1,6; 6,1; 2,5; 5,2; 3,4; and 4,3. Thus, 6/36 = 1/6.

There are two outcomes in each trial of this experiment, and there are ten total trials. Thus, 2 raised to the tenth power yields an answer of 1024.

The selection of the second card is an independent event because it is unaffected by the first event.  If the king of hearts had not been replaced, then the probability of selecting a particular card would have been affected by the first event, and the second selection would have been dependent.  This, however, is not the case in this question.

The sample space, or total possible outcomes, when rolling two six-sided dice is .

Ways to get what you want:

So there are ways to get a .

So the probability becomes

Two events are independent of each other when the result of one does not affect the result of the other.  In the case of the coin being flipped, the first result in no way influenced the result of the second coin flip.  In contrast, when a card is removed from a deck of cards and set aside, that card cannot be selected when a second card is taken from the deck. 

Because the two events are known to be independent, then the following is true by definition.

.

This then becomes an algebra problem:

Because the two are independent, the calculation becomes the product of the two by definition.

We need to recall that  respresents the compliment of A which is everything that is not in A or in mathematical terms:

.

Likewise for the compliment of B:

Therefore to find the intersection of these two independent events we multiply them together.

The gender of each child can be considered an independent event. Each child has a 50% chance of being a boy, and whether a boy was already born previously does not affect the next child's gender.

These two events are independent of one another. During sampling with replacement, the first card does not affect the second card being picked.

To illustrate, consider the probability of drawing a heart first

Assuming you first drew a heart and replaced it in the deck, does the probability of drawing a heart as the second card change?

The probability remains the same, there are still 13 hearts and 52 total cards.

These events are not independent, because if one event happens, it affects the probability of the other event happening. Consider the probability of drawing a heart and the probability of getting a heart given a heart was already drawn. If these two probabilities are the same, the events are independent. If the two probabilities are not the asme, the events are not independent.

After a heart has already been drawn, there are now only 52 cards total and 12 hearts left. These two probabilities are not equal, therefore the events are not independent.