The easiest way to determine the probability of at least one head is to determine the complement of this event, which is the probability of two tails. The penny will come up tails 35% of the time (0.35) and the nickel will come up tails 40% of the time (0.40). These are independent events, so the probability of both happening is the product, or

.

The probability of at least one head - the complement of this event - is this probability subtracted from one, so

, or 86%.

For a roll of "11" to occur with two dice, one die must show a "5" and the other must show a "6". In the fair die, "5" and "6" are two of six equally likely outcomes, so each will happen with probability . 

In the loaded die, since a "6" will come up with probability , for the other five rolls to be equally likely, each, including "5", must come up with probability 

.

The probability of rolling a "5" on the loaded die and a "6" on the fair die is

.

The probability of rolling a "6" on the loaded die and a "5" on the fair die is

.

Add these probabilities:

For a "2" to be rolled with two dice, both dice must show a "1".

For each die, since a "6" will come up with probability , for the other five rolls to be equally likely, each, including "1", must come up with probability 

.

The probability of a double "1" showing up will be

.

For a "2" to be rolled with two dice, both dice must show a "1". In the fair die, this is one of six equally likely outcomes, so it will happen with probability . 

In the loaded die, since a "6" will come up with probability , for the other five rolls to be equally likely, each, including "1", must come up with probability 

.

A double "1" will appear with probability

.

A fair dice has 6 faces.  Therefore, there is a sixth chance to land on any number between 1 through 6.  Rolling the die the first time is independent from the second time, which means that we must account for the probabilities for each trial.

Probability of rolling a 6 is:  1/6

Since probability adds up to 1, the odds of rolling anything other than a 6 for one trial is:

1 - 1/6 = 5/6

On two trials:  

To find the probability of an event, we will use the following formula:

Given the event of drawing a red card from a deck of cards, we can calculate the following:

because there are 26 red cards in a deck

•          13 hearts
•          13 diamonds

Now, we can calculate the following:

because there are 52 total cards we could potentially draw.

So, we can substitute. We get

Therefore, the probability of drawing a red card from a deck of cards is .

To find the probability of an event, we will use the following formula:

Now, given the event of a blue car leaves next, we can calculate the following:

because there are 3 blue cars in the lot.

We can also calculate the following:

because there are 9 total cars in the lot that could potentially leave:

•          3 blue cars
•          2 white cars
•          4 black cars

Now, we can substitute. We get

To find the probability of an event, we will use the following formula:

So, given the event of drawing a black card from a deck of cards, we can calculate the following:

because there are 26 black cards in a deck

•          13 clubs
•          13 spades

Now, we can calculate the following:

because there are 52 total cards in a deck we could potentially draw. So, we can substitute. We get

Therefore, the probability of drawing a black card from a deck of cards is .

To find the probability of an event, we will use the following formula:

Now, given the event of selecting a history book, we can calculate the following:

because there are 3 history books on the shelf.

We can also calculate the following:

because there are 18 total books on the shelf we could potentially choose from:

•          10 chapter books
•          5 non-fiction books
•          3 history books

Now, we can substitute.  We get

Therefore, the probability of selecting a history book is .

To find the probability of an event, we will use the following formula:

Now, we will look at finding the probability of drawing a King, Queen, or Jack from a deck of cards. We will consider these as 3 separate events. We will find the probability of each, then add them together.

So, we will find the probability of drawing a King from a deck of cards. We get

because there are 4 Kings in a deck of cards:

•          King of Hearts
•          King of Diamonds
•          King of Clubs
•          King of Spades

We also get

because there are 52 cards in a deck that we would potentially draw. 

So, we get

Now, we will do the same for finding the probability of drawing a Queen from a deck of cards.

because there are 4 Queens in a deck of cards:

•          Queen of Hearts
•          Queen of Diamonds
•          Queen of Clubs
•          Queen of Spades

We also get

because there are 52 cards in a deck that we would potentially draw. 

So, we get

And lastly, we will find the probability of drawing a Jack.  We get

because there are 4 Jacks in a deck of cards:

•          Jack of Hearts
•          Jack of Diamonds
•          Jack of Clubs
•          Jack of Spades

We also get

because there are 52 cards in a deck that we would potentially draw.  

So, we get

Now, to find the probability of drawing a King, Queen, or Jack, we will add them all together.  We get

Therefore, the probability of drawing a King, Queen, or Jack from a deck of cards is .