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The probability that it will rain today is $0.5$. The probability that it will rain tomorrow is $0.6$. The probability that it will rain either today or tomorrow is $0.7$. What is the probability that it will rain today and tomorrow?

1. $0.3$
2. $0.25$
3. $0.35$
4. $0.4$
edited | 1.7k views
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Inclusion-Exclusion Principle (2 Sets)

Answer: $D$

$P$(it will rain today either today or tomorrow) = $P$(it will rain today) $+ P$(it will rain tomorrow) $- P$(it will rain today and tomorrow)

So, $0.7 = 0.5 + 0.6 - P$(it will rain today and tomorrow)

$\Rightarrow P$ (it will rain today and tomorrow) =$0.4$
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Here Events are independent events .If A is event :"Rains Today" and B is event :"Rains Tomorrow" then Raining tomorrow doesn't depend on raining today.So A and B are independent events and for independent events P(A and B ) is given by P(A)* P(B).

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@vivek  ... I agree with you. But then The probability that it will rain either today or tomorrow should be - It rains today but not tomorrow + It does not rain today but rains tomorrow = (0.5) * (0.4) + (0.5) * (0.6) =( 0.5 ) , but in the question it is given 0.7

Can any one please make it clear ... ?

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@Vijay - yes, and that shows that the events are not independent. If it rains today, the probability of it raining tomorrow increases- may be showing that it is Monsoon season :)
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Sir... What if The probability that it will rain either today or tomorrow is (0.7) not given ??

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In that case we cant say the events are dependent or otherwise and usually have to consider real world scenario. I would say consider "events as independent" is its surely the case - as throwing dice 2 times, or if we need to use the independent event formula- no other data given.
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@vivek  ... I agree with you. But then The probability that it will rain either today or tomorrow should be - It rains today but not tomorrow + It does not rain today but rains tomorrow = (0.5) * (0.4) + (0.5) * (0.6) =( 0.5 ) , but in the question it is given 0.7

If we consider Events are independent then

P(A U B) = P(A) + P(B) - P(A)*P(B)

P(it will rain today either today or tomorrow) = P(it will rain today) +P(it will rain tomorrow) −P(it will rain today)*P(it will rain tomorrow) ==> 0.8

how you got 0.5 ..where i am wrong above ??

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@jatin khachane 1

You are correct, it will be 0.8 only.

@vijaycs has calculated it wrongly for $p(A \oplus B) = p(A \cap \overline{B}) + p(B \cap \overline{A} ) - 2p(A \cap B)$

0.7 = 0.5 + 0.6 - p(today and tomorrow) = 0.40
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