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I have this question on an ungraded practice test.

The possible outcomes of an experiment are 1,2,3,4,5,6,7,8, and assume that all of the outcomes are equally likely. Let A be the event that the result is odd, and let B be the event that the result is less than 4. Calculate P(A and B). Round your answer to 3 decimal places.

Here's how I went about solving it:

P(A) = 4/8 = 1/2 P(B) = 3/8

P(A and B) = P(A)*P(B) = (1/2) * (3/8) = 3/16

However the correct answer is 0.25. Why is my answer wrong?

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    $\begingroup$ Welcome to the site! It's because P(A and B) = P(A)P(B) works only when A and B are independent, are you familiar with that term? $\endgroup$
    – John Madden
    Commented Sep 13, 2022 at 12:44
  • $\begingroup$ Yes, that's when the probability of two events is not dependent on each other. $\endgroup$
    – Cat
    Commented Sep 13, 2022 at 12:46
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    $\begingroup$ You cannot use formula $P(A \text{ and } B) = P(A)P(B)$ unless you first prove that $A$ and $B$ are independent events. In this case, it turns out that they are not independent. The simplest way to answer is to explicitly remark that $A = \{1,3,5,7\}$ and $B = \{1,2,3,4\}$, and thus $A \text{ and } B = A \cap B = \{1,3\}$; so $P(A \cap B) = 2 / 8 = 0.25.$ $\endgroup$
    – Stef
    Commented Sep 13, 2022 at 13:17
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    $\begingroup$ How is this a question about union at all? Please edit the title of your question. $\endgroup$
    – Dilip Sarwate
    Commented Sep 13, 2022 at 13:49
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    $\begingroup$ Yes: check the definition. See stats.stackexchange.com/questions/303753. In your case you could draw a diagram of the sample space like this: $$\begin{array}{cccc} 1\ & 3\ \mid & 5 & 7 \\ \hline 2\ \mid & 4 & 6 & 8 \end{array}$$ There are four odd numbers with probability $4/8,$ three numbers less than $4$ (with probability $3/8$) and two in both sets (which is called their intersection) with probability $2/8.$ All you have to do is check whether $4/8\times 3/8=2/8.$ $\endgroup$
    – whuber ♦
    Commented Sep 13, 2022 at 16:23

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