By definition, a discrete random variable is a random variable whose values can be "counted" one by one. A continuous random variable is a random variable that can take any value on a certain interval. Of these choices, the number of lip products, the amount of money, and the number of midterms taken are all discrete random variables, as the respective values can be counted; however, the time taken to watch the first four seasons of a TV show is a continuous random variable, as not everyone will take the same amount of time to watch all those episodes (i.e. some might fastf-orward/replay parts of episodes).

This question pertains to the concept of Geometric Probability Distribution, which states that the probability of  trials until the first success is as follows:

In this equation,  is the probability of success (in this case, a driver caught texting), and  is the probability of failure.

In this particular problem, , and . To calculate the probability of finding that the first driver caught texting is in the thirteenth car stopped, we can calculate . The final answer is .

Two steps are crucial here.

First, we need to recognize this is a binomial distribution with  and .

Second, we need to realize we can use a normal approximation of the binomial since  and , which are both larger than 5.

With that said, we can calculate a -score and its -value, keeping in mind that our mean will be  and our standard deviation will be , which is about .

To calculate Prob(at least one bull's eye), we can instead compute one minus the complementary probability, P(no bull's eye).

So we have P(at least one bull's eye)=1-P(no bull's eye).

The chance of getting no bull's eyes is .

This means the probability of getting at least one bull's eye is 

The movements that this particle can make include: RRL, RLR, LRR.

The chance of getting RRL is . This is also the chance of getting any of those movements.

To get the total probability, we can add up the individual probabilities since the events are all mutually exclusive.

Thus, we get the following as the solution.

To calculate this probability, we need to calculate the chance of getting 4 tails and then a head.

Each tail has a prob. of  and a head is , so we multiply  to the power of 4 (because we need 4 tails) by  (for the single head).

So the probability is 

.

There are four conditions that need to be satisfied for a binomial experiment:

1) Each trial must have two outcomes.

2) Each trial must be independent.

3) All trials must be identical.

4) The probabilities of the outcomes remain constant must not change with each trial.

The only choice that satisfies all four of these conditions (and is therefore a binomial experiment) is the rock-paper-scissors scenario.

A discrete variable is a variable which can only take a countable number of values. For example, the number of times that a coin can come up heads in ten flips can only be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10. Thus, there are a countable number of possible outcomes (in this case 11). This is true for coin flips, but not for caterpillar length, water flow, or rates of return for stocks.

The answer is 17.5. Simply take the values for each day, add them, and divide by the total number of days to obtain the mean: 

We are required to find the mean outcome where the probability of each possible result varies--the random/weighted mean. 

First, multiply each possible outcome by the probability of that outcome occurring. 

Second, add these results together.