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Joint Entrance Examination
Mathematics
Probability
Question 191
Question 191 of 417
Medium
MCQ
A
,
B
A,B
A
,
B
independent,
P
(
A
)
=
p
P(A)=p
P
(
A
)
=
p
,
P
(
B
)
=
2
p
P(B)=2p
P
(
B
)
=
2
p
. Find the largest
p
p
p
for which
P
(
exactly one of
A
,
B
)
=
5
9
P(\text{exactly one of }A,B)=\dfrac59
P
(
exactly one of
A
,
B
)
=
9
5
.
A
1
/
3
1/3
1/3
B
2
/
9
2/9
2/9
C
4
/
9
4/9
4/9
D
5
/
12
5/12
5/12
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Official solution
\[p(1-2p)+(1-p)2p = \frac59 \implies 36p^2-27p+5=0 \implies (3p-1)(12p-5)=0\] \[p=\frac13\text{ or }\frac{5}{12} \implies p_{\max}=\frac{5}{12}\]
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