What is the probability of drawing a red marble from a box containing blue marbles?

You are correct. Another way to compute it:

Let $A_i$ be the event in which the $i$-th ball you grab is red. For the first ball you choose, you have $3$ red marbles out of a total of $18$. Therefore,

$$P(A_1) = \frac{3}{18}.$$

For the second marble you grab, given that you already took a red one, that is, given $A_1$, you have $2$ red balls out of a total of $17$, then

$$P(A_2 \mid A_1) = \frac{2}{17}.$$

Finally,

$$P(A_1 \cap A_2) = P(A_1)P(A_2 \mid A_1) = \frac{3}{18}\frac{2}{17} = \frac{1}{51}.$$

You can use the same reasoning to compute for the purple ones and then add those probabilities.

Video transcript

Find the probability of pulling a yellow marble from a bag with 3 yellow, 2 red, 2 green, and 1 blue-- I'm assuming-- marbles. So they say the probability-- I'll just say p for probability. The probability of picking a yellow marble. And so this is sometimes the event in question, right over here, is picking the yellow marble. I'll even write down the word "picking." And when you say probability, it's really just a way of measuring the likelihood that something is going to happen. And the way we're going to think about it is how many of the outcomes from this trial, from this picking a marble out of a bag, how many meet our constraints, satisfy this event? And how many possible outcomes are there? So let me write the possible outcomes right over here, so possible outcomes. And you'll see it's actually a very straightforward idea. But I'll just make sure that we understand all the words that people might say, so the set of all the possible outcomes. Well, there's three yellow marbles. So I could pick that yellow marble, that yellow marble, or that yellow marble, that yellow marble. These are clearly all yellow. There's two red marbles in the bag. So I could pick that red marble or that red marble. There's two green marbles in the bag. So I could pick that green marble or that green marble. And then there's one blue marble in the bag. There's one blue marble. So this is all the possible outcomes. And sometimes this is referred to as the sample space, the set of all the possible outcomes. Fancy word for just a simple idea, that the sample space, when I pick something out of the bag, and that picking out of the bag is called a trial, there's 8 possible things I can do. So when I think about the probability of picking a yellow marble, I want to think about, well, what are all of the possibilities? Well, there's 8 possibilities, 8 possibilities for my trial. So the number of outcomes, number of possible outcomes, you could view it as the size of the sample space, number of possible outcomes, And it's as simple as saying, look, I have 8 marbles. And then you say, well, how many of those marbles meet my constraint, that satisfy this event here? Well, there's 3 marbles that satisfy my event. There's 3 outcomes that will allow this event to occur, I guess is one way to say it. So there's 3 right over here, so number that satisfy the event or the constraint right over here. So it's very simple ideas. Many times the words make them more complicated than they need to. If I say, what's the probability of picking a yellow marble? Well, how many different types of marbles can I pick? Well, there's 8 different marbles I could pick. And then how many of them are yellow? Well, there's 3 of them that are actually yellow.

We have seen that it is possible to find the probability of compound events, where we have the occurrence of more than one simple event in a sequence. When working with more than one event, you have to be concerned as to whether the first event affects the second event. When determining if events are independent, you are determining if the events are affecting one another. Two events are said to be independent if the result of the second event is not affected by the result of the first event. The probability of one event does not change the probability of the other event. If A and B are independent events, the probability of both events occurring is the product of the probabilities of the individual events. If A and B are independent events, P(A∩B) = P(A and B) = P(A) • P(B). (referred to as the "Probability Multiplication Rule") Notice the connection between "AND" and "multiplication".   What is the probability of tossing a head on a penny and then choosing an ace from a standard deck of cards? These are independent events as the second event is not affected by the first. The probability of BOTH of these events is found by the Multiplication Rule.  The events are independent. P(head then ace) = P(head) • P(ace) = 1/2 • 4/52 = 2/52 = 1/26. A drawer contains 3 red paper clips, 4 green paper clips, and 5 blue paper clips.  One paper clip is taken from the drawer and then replaced.  Another paper clip is taken from the drawer.  What is the probability that the first paper clip is red and the second paper clip is blue? Because the first paper clip is replaced, the sample space of 12 paper clips does not change from the first event to the second event.  The events are independent. P(red then blue) = P(red) • P(blue) = 3/12 • 5/12 = 15/144 = 5/48. When you toss a coin, the probability of getting a head is 1 out of 2 or ½. If you toss the coin again, the probability of getting a head is still 1 out of 2 or ½. If you toss a coin 10 times and get a head each time, you may think that your luck of tossing a tail is increasing since it has not yet appeared. This is not the case. These events are independent events and do not affect one another. The probability of tossing a tail is 1 out of 2 or ½ regardless of how many heads were tossed previously. Dependent Events (Not independent) If the result of one event IS affected by the result of another event, the events are said to be dependent, or not independent. If A and B are dependent events, the probability of both events occurring is the product of the probability of the first event and the probability of the second event once the first event has occurred. If A and B are dependent events, and A occurs first, P(A and B) = P(A) • P(B, once A has occurred) ... and is written as ... P(A∩B) = P(A and B) = P(A) • P(B | A) The notation P(B | A) is called a "conditional probability" and is read "the probability of event B given that event A has occurred".   A bag contains 3 green marbles and 2 red marbles. A marble is drawn, not replaced, and then a second marble is drawn. What is the probability of drawing a green marble followed by drawing a red marble? By not replacing the marble after the first draw, the probability of the second draw is affected. The sample space of the second draw has changed, leaving only 4 marbles. The events are dependent. P(green then red) = P(green) • P(red given green occurred) = 3/5 • 2/4 = 6/20 = 3/10.   A drawer contains 3 red paper clips, 4 green paper clips, and 5 blue paper clips.  One paper clip is taken from the drawer and is NOT replaced. Another paper clip is taken from the drawer.  What is the probability that the first paper clip is red and the second paper clip is blue? Because the first paper clip is NOT replaced, the sample space of the second event is changed.  The sample space of the first event is 12 paper clips, but the sample space of the second event is now 11 paper clips.  The events are dependent. P(red then blue) = P(red) • P(blue given red occurred) = 3/12 • 5/11 = 15/132 = 5/44. Sampling with, and without, replacement: When working with the probability of two (or more) events occurring, it is important to determine if finding the probability of one of the events has an effect on any of the other events. Consider the following example: What is the probability of drawing a red marble, then drawing a blue marble from this jar? The probability of drawing a red marble = 2/5. The probability of drawing a blue marble = 1/5. BUT... • The 1/5 probability of drawing a blue marble assumes all 5 marbles are in the jar. • What happens if you draw the first marble and do NOT put that marble back in the jar before drawing the second marble? If the marble is not "replaced", the probability of the second drawing changes, since there are less marbles in the jar. The probability of drawing a red marble = 2/5. The probability of drawing a blue marble is now = 1/4. Let's compare the two different answers: With Replacement: Without Replacement: The probability of drawing a red marble = 2/5. Put the marble back in the jar. The probability of drawing a blue marble = 1/5. (of the 5 in the jar) Answer: 2/5 • 1/5 = 2/25 The probability of drawing a red marble = 2/5. Do not put marble back in jar. The probability of drawing a blue marble = 1/4. (of the 4 left in jar) Answer: 2/5 • 1/4 = 2/20 = 1/10 In relation to probability, the word "replacement" most often refers to situations where something can be "removed" (drawn, chosen, etc.) from the sample set, and then replaced (or not replaced). • " With replacement": Choosing a ball, a card, a marble, or other object, and then replacing the item back into the sample space each time an event occurs. Example: Choosing a card from a deck and then putting the card back into the deck before drawing another card. • "Without replacement": Choosing a ball, a card, a marble, or other object, and then NOT replacing the item back into the sample space before choosing another object. Example: Choosing a card from a deck and not replacing it to the deck before drawing another card. The sample space for the second card draw has now been changed to one less card. Be on the lookout for the word "replacement" as a clue. With Replacement: the events are independent. Probabilities do NOT affect one another. Without Replacement: the events are dependent. Probabilities DO affect one another. NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the "Terms of Use".

What is the probability of drawing a red or blue marble?

What is the probability that it is a red marble?

What is the probability of drawing a red marble out of the bag?

What is the probability of drawing a red marble replacing it then drawing a blue marble from this jar?