JEE Main Math Class 12 Probability Part 1 Questions

Given, B1,B2 and B3 are three independent events.
Let x,y,z be the probability of B1,B2,B3′ respectively.
‌ P(only ‌B1‌ occur ‌)‌‌=α
P(B1)⋅P(B2)⋅P(B3)‌‌=α
⇒‌‌x⋅(1−y)⋅(1−z)‌‌=α
⇒‌ P(only ‌B2‌ occur ‌)=β
P(B1)⋅P(B2)⋅P(B3)‌‌=β
⇒‌‌(1−x)⋅y⋅(1−z)‌‌=β
P(none occur) =P
‌P(B1)⋅P(B2)⋅P(B3)=P
⇒(1−x)⋅(1−y)⋅(1−z)=P
Now, we have given relations
(α−2β)P=αβ
⇒[x(1−y)(1−z)−2y(1−x)(1−z)]
(1−x)(1−y)(1−z)
=x⋅(1−y)(1−z)⋅y(1−x)(1−z)
‌‌[ putting the value of α,β,P]
⇒‌‌(1−z)[x(1−y)−2y(1−x)]=x⋅y⋅(1−z)
⇒‌‌x−xy−2y+2xy=xy
⇒‌‌‌‌x=2y . . . (i)
Similarly, on solving the second relation,
(β−3γ)P=2βγ by putting β,γ and P,
We get, y=3z . . . (ii)
From Eqs. (i) and (ii), we get
x‌‌=2×3z⇒x=6z
⇒‌‌‌

x

z

‌‌=6
Now, ‌

P(B1)

P(B3)

=‌

x

z

=6
∴ Required ratio is 6 .