for which of the following probability assignments are events a and b independent?

 Alright, let's talk about independence when it comes to probability assignments. When events A and B are independent, it means that the occurrence of one event doesn't affect the probability of the other event happening. In simple terms, they're like two peas in a pod, and what one does doesn't bother the other.

Now, to check for independence, we need to look at the probability assignments. You've got some options there, buddy, so let's break it down:

  1. P(A) = 0.4 and P(B) = 0.6
  2. P(A) = 0.2 and P(B) = 0.2
  3. P(A) = 0.3 and P(B) = 0.3
  4. P(A) = 0.1 and P(B) = 0.9

To figure out if A and B are independent, you gotta do a little math. If P(A) multiplied by P(B) equals the probability of both A and B happening (P(A ∩ B)), then they're independent. So, let's do the math for each option:

  1. P(A) * P(B) = 0.4 * 0.6 = 0.24, and P(A ∩ B) isn't given. So, can't say for sure here.

  2. P(A) * P(B) = 0.2 * 0.2 = 0.04, and P(A ∩ B) isn't given. So, again, can't determine independence.

  3. P(A) * P(B) = 0.3 * 0.3 = 0.09, and once more, P(A ∩ B) isn't provided. Can't make the call.

  4. P(A) * P(B) = 0.1 * 0.9 = 0.09, and here P(A ∩ B) isn't given either. So, no independence verdict.

Looks like we can't confidently say if events A and B are independent based on the information provided for any of these options. To check for independence, we need to know both the individual probabilities (P(A) and P(B)) and the probability of both events happening together (P(A ∩ B)). Without that, it's like trying to solve a jigsaw puzzle with missing pieces.

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