The replacement of one card by the joker in a deck of 52 decreases the number of cards of the color of the removed card by 1, leaves as is the number of cards of the other color, and leaves as is the total number of cards. Therefore, the probability that a randomly drawn card will be of the color of the removed card will be reduced. and the probability that it will be of the other color will be the same. 

From Statement 1 alone, since the probability of drawing a red card from the altered deck is the same as that of drawing a red card from an unaltered deck, it follows that the removed card is black. From Statement 2 alone, since the probability of drawing a black card from the altered deck has been reduced, it follows again that the removed card is black. 

Assume Statement 1 alone. There were 26 red cards out of 52 total before the switch, and, since the replacement card was of the same color as the removed card, there were 26 red cards out of 52 after the switch. The probability of drawing a red card stayed the same.

Assume Statement 2 alone. Again, there were 26 red cards out of 52 total before the switch. However, Statement 2 leaves open the possibility of both cards having the same color or different colors, since both black (clubs) and red (hearts and diamonds) cards could have been removed or added. If both cards have the same color, then as in Statement 1, the probability stays the same. But if, for example, the removed card is a club and the added card is a diamond, there are now 27 red cards out of 52, and the probability of drawing a red card has increased to . This makes Statement 2 alone inconclusive.

The probability of drawing an ace (one of thirteen ranks) from an unaltered deck is . From either statement alone, it can be determined that the altered deck has 53 cards, 4 of which are aces; this makes the probability of drawing an ace from this deck . The probability has decreased.

A guitarist grabs a guitar pick out of a dish at random. Find the odds that the pick is green.

I) There are 3 different colors of picks. There are 15 green picks, 45 red picks, and n blue picks

II) There is a total of 143 picks in the dish

Recall that probability can be found by the following:

II) Gives us the total number of outcomes

I) Gives us the desired number of outcomes

So our answer can be found by doing the following:

So there is about a 10.49% chance of getting a green pick.

Don't be distracted by the "n" number of blue picks. We still need II) to find the total number of picks, so both are needed.

The total number of balls in the box will be 

.

Since 

,

it follows that the number of balls is

 .

The number of balls  with a given letter of the alphabet is equal to the number of its position in the alphabet; the probability of a ball with that letter being drawn is that number divided by the total number of balls, 351. Therefore, for this probability to exceed , we must have the relation

.

Therefore, . 

The 11th letter of the alphabet is "K", so in order to answer this question, it suffices to know whether the letter comes after "K" in the alphabet.

The question cannot be answered from either statement alone; both "Barack" and "Obama" include at least one letter that comes after "K" and at least one that does not. However, if both statements are assumed, since both of the letters shared by the words - "A" and "B" - come before "K", the question can be answered in the negative.

The total number of balls in the box will be 

.

Since 

,

it follows that the number of balls is

 .

The number of balls  with a given letter of the alphabet is equal to the number of its position in the alphabet; the probability of a ball with that letter being drawn is that number divided by the total number of balls, 351. Therefore, for this probability to exceed , we must have the relation

.

Therefore, . 

The 5th letter of the alphabet is "E", so in order to answer this question, it suffices to know whether the letter comes after "E" in the alphabet.

Either statement alone is sufficient to answer this question in the affirmative. Statement 1 establishes that the letter must be "I", "M", "P", or "S", all of which come after "E". Statement 2 establishes that the letter cannot be any of "A", "B", "C", "D", or "E", all five of which appear in the word "carbide".

The total number of balls in the box will be 

.

Since 

,

it follows that the number of balls is

 .

The frequencies out of 55 of each outcome from 1 to 10, in order, are as follows:

Their respective probabilities are their frequencies divided by 55:

.

The probability that the ball will be marked "5" is 

;

therefore, the probability that the ball will be marked with any given integer less than or equal to 5 will be greater than . 

The probability that the ball will be marked "6" is

 ;

therefore, the probability that the ball will be marked with any given integer greater than or equal to 6 will be less than . 

Therefore, it suffices to know whether the number on the ball is less than or equal to 5. Statement 2 states that the number on the ball is less than or equal to 5, so it is sufficient to answer the question in the affirmative. Statement 1 is insufficient, since there are primes less than or equal to 5 - 2, 3, and 5 - and one prime greater than 5, which is 7.

The total number of balls in the box will be 

.

Since 

,

it follows that the number of balls is

 .

The frequencies out of 55 of each outcome from 1 to 10, in order, is as follows:

Their respective probabilities are their frequencies divided by 55:

.

The probability that the ball will be marked "5" is 

;

therefore, the probability that the ball will be marked with any given integer less than or equal to 5 will be greater than . 

The probability that the ball will be marked "6" is

 ;

therefore, the probability that the ball will be marked with any given integer greater than or equal to 6 will be less than . 

Therefore, it suffices to know whether the number on the ball is less than or equal to 5.

Statement 1 alone is insufficient, since there are two perfect square integers from 1 to 5 (1 and 4) and one perfect square integer from 6 to 10 (9). Statement 2 alone is insufficient, since it is not clear whether the number on the ball is 5 or a number greater than 5. However, from the two statements together, it can be inferred that , and that the probability of drawing a ball with this number is .

Probability is a part to whole comparison.  

With statement 1, we don't know nor can we determine the probability of drawing a green marble because we know nothing about the probability of drawing a blue marble. Therefore, statement 1 is not sufficient.

With statement 2, we don't know nor can we determine anything regarding the probability of drawing a red marble. Therefore, statement 2 is not sufficient.

However, taken together, we can determine that the probability of drawing a green marble is . Therefore, BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. 

Probability is the number of desired outcomes divided by the number of possible outcomes. 

With statement 1, we are told the probability of never flipping a head is , which is the same as always flipping a tail.  Since the probability of flipping a tail on a single flip is , we can determine the probability of flipping n tails is , which solves as  .  With n=3, we can solve for the probability of getting at least one tail is 1-probability of never getting a tail or 1-HHH:

.  

Therefore, statement 1 alone is sufficient.

With statement 2, we are told the probability of flipping at least one head is  . Since there are only 2 possible outcomes with each flip (heads or tails), the probability of flipping at least one head is the same as the probability of flipping at least one tail. Therefore, statement 2 alone is sufficient.

Therefore, the correct answer is EACH statement ALONE is sufficient.