This is a binomial probability experiment. 

The probability that a fair coin will come up heads in any single toss is . 

If the coin is tossed four times, then the probability that heads will appear three times is:

The probability that heads will appear four times is:

So the probability that heads will appear at least three times is 

The second statement is irrelevant; the result of a previous trial has no influence on that of the current trial. It has already been established in the problem that the coin is fair.

The answer is that Statement 1 alone is sufficient to answer the question, but not Statement 2.

We need both statements to get the answer.

Statement 1 alone is not enough. Having 24 total cookies does not tell us the proportions of each type of cookie. We could have 1 chocolate chip cookie and 23 oatmeal raisin, or vice versa. 

Statement 2 alone is not sufficient either. Knowing that there are 8 more chocolate chip cookies than oatmeal raisin without knowing the total amount in the bag does not help. It could be 1 oatmeal raisin cookie and 9 chocolate chip, or it could be 100 oatmeal raisin cookies and 108 chocolate chip cookies (although that is a pretty big bag!)

Only by having both statements together can we find the right proportion. If we let  be the number of chocolate chip cookies, and  be the number of oatmeal raisin, we can set two equations and find the right ratio. We know that

from statement 1, and

from statement 2.

Substituting  into the first equation we get

 Therefore, we get 8 oatmeal raisin cookies, and 16 chocolate chip cookies. The answer is

or 66.67% chance that the monster pulls out a chocolate chip cookie. 

Pizza received 24 votes from men. Since half of those who responded pizza were women, there must be 24 votes from women.

If 1 out of 5 men voted for pizza, this means we have  men.  If 3 out of 10 women voted for pizza, we have women for a total of 200 people polled.

If our population is instead given as 60% male and 40% female, we still have 24 men and 24 women that voted for pizza, but while we can determine the ratio of people who voted, we cannot get a precise number.  For example, while 80 women and 120 men works, we could also have 160 women and 240 men as our total poll (with 1 out of 10 men and 3 out of 20 women voting pizza). 

Thus the first condition is sufficient while the second is not.

This question deals with conditional probability.

Statement 1) consists of independent events. The past trials will not affect the future trials of the coin.  The odds of flipping a coin and land on heads will be .

Statement 2) also consists of independent events. However, given that the previous 10 days were sunny does not necessarily indicate that the next day will guarantee that it will be a sunny day.  The probability is unknown because the statement does not give any information about the percentage of a sunny sky on any particular day.

To find the probability of several events, we need to find the probability of each event separately. First, we need to know how many there are of each color pick.

Statement I relates the number of orange and blue picks. 

Statement II tells us the number of black picks.

From Statement I:

From Statement II:

Using Statement I, Statement II, and the given info, we can substitute  from Statement I into Statement II for  and get the following:

We can then solve for :

Since we are told in Statement I that there are 14 more blue picks than orange picks, we can then find the number of orange picks:

So, to find the odds of multiple events, multiply the probability of each event together:

Each statement alone gives information about only one order, so at the very least, both statements will be necessary.

From Statement 1, it is known that Ralph will pay  for his drinks, since he will only pay for the two espressos. 

From Statement 2, it cannot be determined exactly what Janet will pay. However, she will pay for all three drinks, two of which are cappucinos; the least she will spend is the price of two cappucinos and an Americano, which is .

From both statements together, it is definite that Janet will pay more.

From Statement 1 alone, you know that Patty spent . It is possible, however, that Mickey spent more (for three espressos, for example), just as much (on three Turkish coffees), or less (for three cappucinos, for example). Statement 2 alone gives you even more scant information, since it only tells you one beverage Mickey bought.

Suppose you know both statements. Then you know Patty spent . You do not know how much Mickey spent, but the most he could have spent was  (for an Americano and two esperessos). Therefore, the two statements together are necessary and sufficient to answer the question.

 students voted, so 236 votes were necessary to win a simple majority. 

From Statement 1 alone it can be determined that Crane won at least  votes (his own initial votes plus all of the votes that previously went to Jones), and Trask won at least 98 (his own initial votes).

From Statement 2 alone, it can be determined that Crane won at least  (his own initial votes plus one-third of the votes that previously went to Wells), and Trask won at least  votes (his own initial votes plus two-thirds of the votes that previously went to Wells). 

Neither statement alone will prove a majority vote. From Statement 1 and 2 together, however, it is known that Crane won at least his own initial votes, all of Jones' votes, and one-third of Wells' votes - a minimum of 

 votes. Therefore Crane is the winner, and both statements are needed to prove this.

Statement 1 alone tells us that the population of Renfrow in 1945 was between 13,251 and 15,049, but says nothing about the population in 1955. 

Similarly, Statement 2 alone tells us that the population in 1955 was between 15,049 and 19,415, but says nothing about the population in 1945.

The two statements together, however, tell us that the population rose every year from 1940 to 1960, so the population in 1955 had to have been greater than the population in 1945.

From the two statements together, it can only be surmised that the population in both 1965 and 1975 exceeded the 1970 population. No conclusions can be drawn about the size of the 1965 population relative to the size of the 1975 population.