Choose a number B at random from the interval 0 1 with unifo

Choose a number B at random from the interval [0, 1] with uniform density.

Find the probability that

(a) 1/3<B<2/3.

(b) |B1/2|1/4.

(c) B<1/4 or 1B<1/4.

(d) 3B^2 < B.

please explain in detail. I am facing diffculties to understand this question.

Solution

Note that for uniform distribution

P = (b - a) / L

where

b = upper bound
a = lower bound
L = the length of the distribution = 1 - 0 = 1

*******************

a)

P = (2/3 - 1/3)/1 = 1/3 = 0.33333333 [answer]

*******************

b)

|B - 1/2| <= 1/4

Thus, this means

-1/4 < B - 1/2 < 1/4

1/4 < B < 3/4

Thus,

P = (3/4 - 1/4)/1

P = 1/2 = 0.5 [answer]

*****************

B < 1/4 OR 1 - B < 1/4

B < 1/4 OR B > 3/4

For B < 1/4, the lower bound is 0.

For B > 3/4, the upper bound is 1.

Thus,

P = (1/4 - 0)/1 + (1 - 3/4)/1

P = 1/4 + 1/4

P = 0.5 [answer]

*************************

3B^2 < B

As B is positive anyway, we divide both side by B,

3B < 1

B < 1/3

Here, the lower bound is 0,

P = (1/3 - 0) / 1

P = 1/3 = 0.333333333 [answer]