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I am a beginner, learning probability and statistics.

Yesterday I learned the difference between mutually exclusive (disjoint) events and Independent events (although I'm confusing at times).

Anyway I was solving a question that asked to find the probability when a coin is flipped and if "head" appears then a dice is rolled to get 4.

Now, I was interested to find if coin tossed and rolling a dice are independent or dependent.

Q1: Can I intuitively say that since the two events have nothing in common (like how can flipping a coin and rolling a dice are related) that they are independent?

So, I tried to prove it, but got stuck.

A: coin flipped

B: dice rolled

$P(B|A) = \frac{P(A \cap B)}{P(A)}$

Q2: Now I am not sure on how to define $P(A \cap B).$

Is it probability of having a head AND rolling a dice with head 4?

If so, then $P(A\cap B) = P(A).P(B) = \frac{1}{2}*\frac{1}{6}$

and $P(B|A) = \frac{\frac{1}{2}*\frac{1}{6}}{\frac{1}{2}} = \frac{1}{6} = P(B)$

Then we can say $P(B|A) = P(B)$ and hence independent.

Is this correct?

Q3: Again I have read that mutually exclusive events are events which cannot happen at the same time. Then how can A and B happen at the same time? Then they must be mutually exclusive and cannot be independent.

N.B: Please excuse me if I'm jumbling things up and asking stupid things. I learnt them and now I'm tangled in my own explorations. Seeking your kind help here. Thanks in advance.

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    $\begingroup$ You kind of went circular with that definition because $P(A \cap B) = P(A) \times P(B)$ only when the events are independent, so the logic didn't work. Mutually exclusive events are better thought of as if one happens, then the other can't (so disjoint within the same sample space). So rolling a dice to get a 4 and tossing a coin heads up aren't mutually exclusive. But they are independent because doing one doesn't affect the other. Also note that if events aren't independent, you find $P(A \cap B)$ by $P(A) + P(B) - P(A \cup B)$. $\endgroup$ Jul 2, 2017 at 8:11
  • $\begingroup$ @OsamaGhani : P(head and tail at the same time) = 0, because they are the same entity. P(head and dice with 4 at the same time) = ... because it can happen as dice and coin are two entities? So, they are not mutually exclusive, am I correct? Now how will you find $P(A \cap B)$? I know $P(A)$, $P(B)$ but what is $P(A \cup B)$. Also how can you say "one cannot affect the other" without proving? Is it intuitively? $\endgroup$
    – lu5er
    Jul 2, 2017 at 8:42
  • $\begingroup$ Yes you are right, the dice and coin toss are not mutually exclusive. Finding the probabilities of $P(A \cup B)$ and $P(A \cap B)$ are context dependent. In this case just doing $P(A \cap B) = P(A) \times P(B)$ works. In other cases, you may have to draw tree diagrams, Venn diagrams etc. It really depends on the question. $\endgroup$ Jul 2, 2017 at 8:45
  • $\begingroup$ In which case is that true? For mutually exclusive events, it's true that $P(A \cup B) = 1$. $\endgroup$ Jul 2, 2017 at 8:48
  • $\begingroup$ @OsamaGhani.. Thanks! I think I got it now.. $\endgroup$
    – lu5er
    Jul 2, 2017 at 8:50

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