Two dice are thrown simultaneously. probability that sum is odd or less than 7 or both, is

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14.Probability

medium

Two dice are thrown simultaneously. The probability that sum is odd or less than $7$ or both, is

$\frac{2}{3}$

$\frac{1}{2}$

$\frac{3}{4}$

$\frac{1}{3}$

Solution

(c) Required probability
$ = P({\rm{less}}\,\,{\rm{than}}\,\,7) + P({\rm{odd}}) + P({\rm{both}}) – P(7 \cap {\rm{odd}})$
$ – P(7 \cap {\rm{both}}) – P({\rm{odd}} \cap {\rm{both}}) + P({\rm{odd}} \cap 7 \cap {\rm{both}})$
But $P({\rm{both}}) = P(7 \cap {\rm{odd}}) = P(7 \cap {\rm{both}}) = P(o{\rm{dd}} \cap {\rm{both}})$
$ = P({\rm{odd}} \cap 7 \cap {\rm{both}})$
Therefore required probability
$ = P({\rm{Less\, than 7)}} + P({\rm{odd) — }}P({\rm{7 }} \cap {\rm{odd}})$
$P({\rm{odd}}) = \frac{{18}}{{36}} = \frac{1}{2}$
$P({\rm{less}}\,\,{\rm{than}}\,\,7) = \frac{{15}}{{36}} = \frac{5}{{12}},\,\,P({\rm{both}}) = \frac{6}{{36}} = \frac{1}{6}$
Hence required probability $ = \frac{5}{{12}} + \frac{1}{2} – \frac{2}{{12}} = \frac{9}{{12}} = \frac{3}{4}.$

Standard 11

Mathematics

If $A$ and $B$ are two events such that $P(A) = \frac{1}{2}$ and $P(B) = \frac{2}{3},$ then

hard

Let $A$ and $B$ be two events such that $P\,(A) = 0.3$ and $P\,(A \cup B) = 0.8$. If $A$ and $B$ are independent events, then $P(B) = $

easy

(IIT-1990)

If $P\,(A) = 0.4,\,\,P\,(B) = x,\,\,P\,(A \cup B) = 0.7$ and the events $A$ and $B$ are mutually exclusive, then $x = $

easy

One card is drawn at random from a well shuffled deck of $52$ cards. In which of the following cases are the events $\mathrm{E}$ and $\mathrm{F}$ independent ?

$E:$ ' the card drawn is a king and queen '

$F:$ ' the card drawn is a queen or jack '

medium

Let $A$ and $B$ be independent events such that $\mathrm{P}(\mathrm{A})=\mathrm{p}, \mathrm{P}(\mathrm{B})=2 \mathrm{p} .$ The largest value of $\mathrm{p}$, for which $\mathrm{P}$ (exactly one of $\mathrm{A}, \mathrm{B}$ occurs $)=\frac{5}{9}$, is :

hard

(JEE MAIN-2021)