Probability question: What is the difference in: P(red AND blue) and P(red THEN blue)? Are they independent or dependent. Nothing is said about replacing the first.

Ah, probability, the math version of gambling. Let's dive into your question, shall we?

P(red AND blue) represents the probability of randomly drawing both a red and a blue item simultaneously. This implies that you are taking both items out at the same time, without replacing the first.

On the other hand, P(red THEN blue) refers to the probability of drawing a red item first, and then, without replacement, drawing a blue item. Here, the key point is that you are not putting the red item back before drawing the blue one.

As for their independence or dependence, it depends on the context and the nature of the items. If we're talking about a magical box where the probabilities do not change, then they are independent events. However, if we're dealing with a set of items where the chances of getting a red item affect the chances of getting a blue one, then they are dependent events.

Remember, probability can be a tricky subject! But it's not as tricky as trying to juggle 10 flaming torches while riding a unicycle. Trust me, I've tried.

To answer your question, let's define some terms:

P(red AND blue) refers to the probability of both red and blue events occurring simultaneously. In this case, we are interested in knowing the probability of selecting a red object and a blue object from a set of objects.

P(red THEN blue) refers to the probability of the red event occurring first and then the blue event occurring. In this case, we are interested in knowing the probability of selecting a red object first, and then selecting a blue object from the remaining objects, assuming no replacement is made.

Whether these events are independent or dependent depends on whether the occurrence of one event affects the probability of the other event occurring.

If the events are independent, the probability of one event occurring does not affect the likelihood of the other event occurring. In this case, P(red AND blue) is equal to the product of the probabilities of selecting a red object and a blue object.

If the events are dependent, the probability of one event occurring does affect the likelihood of the other event occurring. In this case, P(red AND blue) is different from P(red THEN blue).

However, without more information about the scenario, such as the number of objects, the probabilities of selecting red and blue objects, and whether replacements are made, it is not possible to calculate the exact probabilities or determine if they are independent or dependent.

To calculate the difference between P(red AND blue) and P(red THEN blue), let's break it down:

1. P(red AND blue): This refers to the probability of both events, red and blue, happening simultaneously. To calculate this, you need to know the probability of red and the probability of blue, and then multiply the two probabilities together.

For example, if the probability of red is 0.4 and the probability of blue is 0.3, then you would calculate P(red AND blue) as 0.4 * 0.3 = 0.12.

2. P(red THEN blue): This refers to the probability of red happening first, followed by blue. To calculate this, you need to multiply the probability of red by the probability of blue, given that red has already occurred.

It is important to note that when the question doesn't specify whether the first item is replaced, we assume that it's not replaced. This means that the probability of blue may be affected by the outcome of the red event.

For example, if the probability of red is 0.4 and the probability of blue, given that red has occurred, is 0.2, then you would calculate P(red THEN blue) as 0.4 * 0.2 = 0.08.

Now, let's determine if these events are independent or dependent:

If P(red AND blue) = P(red) * P(blue), it implies that the events are independent. Independent events are those where the occurrence or non-occurrence of one event has no effect on the occurrence or non-occurrence of the other.

If P(red AND blue) = P(red) * P(blue | red), it implies that the events are dependent. Dependent events are those where the occurrence or non-occurrence of one event affects the occurrence or non-occurrence of the other.

In our example, since P(red AND blue) = 0.12 and P(red) * P(blue) = 0.4 * 0.3 = 0.12, these events are independent.

I hope this explanation helps you understand the difference between P(red AND blue) and P(red THEN blue), as well as the concept of independent and dependent events.

Red and blue means they can be picked in any order, and red then blue is in that order.

also you multiply the fractions (probabilities) by each other with and. With then I'm pretty sure you just add them.