We independently throw 10 balls into one of 3 boxes numbered...

We independently throw 10 balls into one of 3 boxes, numbered 1,2 and 3, with probabilities 1/4, 1/2 and 1/4, respectively. What is the probability that box 1 has 2, box 2 has 5 and box 3 has 3 balls? What is the probability that box 1 remains empty?

Transcript

00:01 So in this question, we are told that we're throwing 10 balls into boxes, right? and each throw we make is going to be independent, right? so it doesn't matter if we throw the ball the first time and it lands into like box a.

00:12 The next time we throw the ball, it's going to have the same probability of landing in box a or box b or box c.

00:17 Okay? so unchanged.

00:21 We are told that the probability of our ball landing in box one is equal to 1 over 4.

00:29 Okay? and then we're told that the probability of it landing in box 2 is going to be equal to one half, and then that the probability of it landing in box 3 is equal to 1 over 4.

00:43 Okay? so, since we're throwing 10 balls, we want to find a specific probability, right? so we want to find the probability that box 1, which we can call b1, has 2, and the probability that box two has five, and the probability that box three has three.

01:16 Okay? so this may look like a mouthful or a handful.

01:23 What's important here that we recognize is our multiplication rule of counting.

01:27 Okay? and we're going to be focusing on the one for independent events because they've already told us that every time we throw a ball, it's going to be independent, okay? so each throw is independent.

01:39 So with that in mind, we can say that when we have a ways to do one thing and we have b ways to do another thing, if both of these things are independent, the number of ways we can do both of these things is a times b.

01:52 And this is important because it also applies to our probabilities and tells us that the probability of events a and b, even though you could extend this to and c and d, e, f, etc., the probability of any two independent events is going to be equal to the probability of the first event, multiplied by this probability of the second event, okay, by the probability of b, given that these two are independent, right? so for any two independent events, this holds true.

02:21 This is going to help us out with our question, right? because we know, following this logic, that if we're trying to find this probability right here, it's the same thing as the probability of getting two into box one.

02:34 Right and or we could just start multiplying right so it's going to be the probability of rolling two into box one multiplied by the probability of rolling five into box two multiplied by the probability of rolling or throwing three into box three right that's how we're going to do it so if we're able to find all these individual probabilities we can answer a question just fine okay so um i'll just make some room here i'll keep that...

Question

We independently throw 10 balls into one of 3 boxes, numbered 1,2 and 3, with probabilities 1/4, 1/2 and 1/4, respectively. What is the probability that box 1 has 2, box 2 has 5 and box 3 has 3 balls? What is the probability that box 1 remains empty?

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