Janice has a cooler with bottles of water, cola, and root beer. What are the odds of choosing a water bottle followed by a bottle of root beer?

I) There are 5 times more bottles of root beer than water.

II) There are 75 beverages total in the cooler.

To calculate probabilty, we need to know the number of desired outcomes and the total number of outcomes. 

II) gives us the number of total possible outcomes.

I) gives us a clue about the number of desired outcomes, but it is insufficient  to find the actual number of root beers and waters.

Therefore 

Neither statement is sufficient to answer the question. More information is needed.

Since this problem deals with a probability with two potential outcomes, it is a binomial distribution, and so the probability of an event is given as:

where  is the number of events,  is the number of "successes" (in this case, a "heads" outcome), and  is the probability of success (in this case, fifty percent).

Thus, the probability of three heads in five flips is:

The probability of either of two events happening, the union, is given by the formula:

Because we are not told that drawing a blue and drawing a nine are mutually exclusive events, we must know the probability of the intersection, of drawing a blue nine.

Since this probability question deals with successes and failures, it regards binomial distributions:

Where   is the number of trials,  is the number of 'successes,'  is the probability of exactly  successes in  trials, and  is the probability of a success occuring.

Statement 1 allows for a calculation of  to be used in the probability calculation for the problem.

Statement 2 outright gives a  to calculate the probability asked of in the problem.

Since this probability question deals with successes and failures, it regards binomial distributions:

Where   is the number of trials,  is the number of 'successes,'  is the probability of exactly  successes in  trials, and  is the probability of a success occuring.

The probability of at least one success is the complement of exactly zero successes. Since we're looking at 3 trials, we can find the probability of zero successes to be  and thus the complement, our .

For fewer than three trials, the probability will always be below , but for three or more, it depends on exactly how many trials there are, since the probability increases with increasing number of trials.

Either of these statements can enable finding the probability of Harold sitting in a given seat, i.e. .

Statement 1 outright provides it, while Statement 2 can lead towards it with the relation 

This is a conditional probability problem, which follows the form:

In this case  is the probability of passing the first test,  is the probability of passing the second test,  is then the probability of passing the second test, having passed the first, and  is the probability of having passed both. The probability or percentage of students that passed the first test is necessary information.

The probability of drawing a red card from a standard deck is .

If a red card is removed from a standard deck of fifty-two cards, and one card is chosen at random, the probability of it being red is . If a black card is removed, the probability is . Therefore, to answer the question, the color of the card must be known; the rank is irrelevant to the question. Statement 1 - but not Statement 2 - provides sufficient information.

The probability of drawing a black card from a standard, unaltered deck is .

If a red card is removed from a standard deck of fifty-two cards and replaced by the joker, and one card is chosen at random, all 26 black cards remain in a deck of 52, so the probability of the chosen card being black is still . If a black card is removed and replaced, there are 25 black cards out of 52 left, so the probability is . Therefore, to answer the question, the color of the card must be known. However, the two statements together, if assumed true, leave two possibilities open - the card could be a diamond (red) or a spade (black). The two statements together provide insufficient information.

Assume both statements to be true. The probability of drawing a black card from a standard, unaltered deck is . We examine two cases.

Case 1: The jack of diamonds from the first deck was replaced by the jack of clubs from the other deck. 

Then there are now 27 black cards in a deck of 52. The probability of drawing a black card at random from this deck has increased from  to .

Case 2: The jack of clubs from the first deck was replaced by the jack of diamonds  from the other deck. 

Then there are now 25 black cards in a deck of 52. The probability of drawing a black card at random from this deck has decreased from  to .

In both scenarios, the conditions of both statements were met; the removed card and its replacement were jacks of different suits. But in one scenario, the probability of drawing a black card increased, and in the other, it decreased. The two statements together provide insufficient information.