Removing the hearts from the deck reduces the number of kings in the deck by one. Since each of the five choices involves replacing the thirteen cards with thirteen different cards, thereby keeping the number of cards constant, we want the choice that restores the number of kings to four - that is, the choice that involves adding exactly one king. The only choice that involves this is the choice to add the clubs from another deck.

The number of balls placed in the box will be 

The sum of the first  natural numbers is 

, 

so there are

,

balls in the box.

The five vowels are "A", "E", "I", "O", and "U", which are the first, fifth, ninth, fifteenth, and twenty-first letters of the alphabet. Consequently, there are

balls marked with vowels, and the probability of drawing one of these balls is

.

The number of balls placed in the box will be 

The sum of the first  natural numbers is 

, 

so there are

,

balls in the box.

"M" is the thirteenth letter of the alphabet, so thirteen balls are marked with an "M". The probability of drawing this letter is

The probability that the spinner will stop in a red sector depends on the angles of the different colored sectors, not on the sizes of the sectors. The radii of the sectots is therefore irrelevant.

Each red region is half of a quarter-circle, so the two sectors are, together, one fourth of a circle. The odds against the spinner stopping in a red circle are therefore three fourths to one fourth, or 3 to 1, against.

An unaltered standard deck of 52 cards contains 13 of each of the four suits. Switching a spade and a club from this deck with a spade and a club from another deck yields an altered deck, but with the same number of cards, and with the same number of cards of each suit. No probabilities changed.

Consider first all of the possible ways the men may be arranged, which is

Now, consider all of the ways that Aaron and Gary could be seated beside each other; it may be easier to visualize by drawing it out:

1. A G _ _
2. G A _ _
3. _ A G _
4. _ G A _
5. _ _ A G
6. _ _ G A

As seen, there are six possibilities.

Finally, for each of these cases, Craige and Boone could be seated in one of two ways.

So the probability that Aaron and Gary will be seated beside each other is:

A coin only has two outcomes, heads or tails.  So if the coin is flipped "t" times, "h" of those times are heads and "t-h" of those times are tails.

From here, we use the simple probability formula:

sowe simply create a fraction that has the number of tails-up divied by the total number of flips;

The probability of any given event is given by the equation 

P= [(number of desired events) / (total possible events)] x 100%

In this example, let x represent the number of blue skirts. 

If there's a 25% probability that the skirt Sarah chooses is blue, then the probability of the event can be written like this:

P = (x/12)(100%) = 25%

To solve for x, we can set up a proportion like this:

By cross-multiplication, we get

Therefore, x=3. 

In order to find the probability of the outcomes that Jennifer, James and Kensington flip, you need to multiply the probability of each outcome by each other. 

The probability of the outcome of an event is given by the number of desired outcomes divided by the total number of possible outcomes. 

For this example, since there are three blue scarves, there are three desired outcomes. In total, there are seven possible outcomes—one for each scarf Amelia could pick. 

Therefore, the probability that she will pick a blue scarf is .