# Rule 2G

We can also use our rules to calculate probabilities of combinations without independence. Rule 2G tells us that the probability of drawing an ace, not putting this card back in the deck, and then drawing a king is 4/52 × 4/51 = 16/2,652. But what is the probability of drawing an ace and a king in any order? It is the probability of drawing either an ace or a king and then drawing the other one given that you drew the first one. That probability by Rule 2G is 8/52 × 4/51 = 32/2,652. The difference between this result and the previous one, where the order was specified, shows why we need to de- termine whether we are dealing with permutations or combinations.

Use the rules of probability to calculate these probabilities:

1. What is the probability of rolling a five on one throw of a fair six-sided die?

2. What is the probability of not rolling a five on one throw of a fair six-sided die?

3. If you roll a five on your first throw of a fair six-sided die, what is proba- bility of rolling another five on a second throw of that die?

4. If you roll two fair six-sided dice one time, what are the chances that both of the dice will come up a five?

5. If you roll two fair six-sided dice one time, what are the chances that one or the other (or both) of the dice will come up a five?

6. If you roll two fair six-sided dice one time, what are the chances that one and only one of the dice will come up a five?

7. If you roll two fair six-sided dice one time, what are the chances that at least one of the dice will come up a five?

8. If you roll two fair six-sided dice one time, what are the chances that at least one of the dice will not come up a five?

9. If you roll six fair six-sided dice one time, what are the chances that at least one of the dice will come up a five?

10. If you roll six fair six-sided dice one time, what are the chances that at least one of the dice will not come up a five?