Question

If a real number x is chosen at random in the interval [0, 3] and a real number y is chosen at random in the interval [0, 4 ,] what is the probability that x<y?

Here's what I got - I'm hoping someone can tell me if this is a valid argument.

P(x<y) = P(y>3) + P(y<3 and x<y|y<3) = 1/4 + (3/4)(1/2) = 5/8.

I know 5/8 is the correct answer. Is this the correct method to get there? Thanks.

Answer #1

Yes, the solution given by you is correct. It is a valid argument.

Following the another method that can be used:

Refer to the attached picture for the figure. We are given that 0 ≤ x ≤ 3 and 0 ≤ y ≤ 4

Thus the region x-y when plotted gives you a rectangle with base
= 3 units and height 4 units (as shown in the picture), giving you
a total area = 4*3=12 units^{2}

Now, realize that y=x is a line that passes through (0,0) and (3,3) and divides the above rectangle into a trapezoid (ABCD) and a triangle. The area y>x will belong to the trapezoidal area.

Thus, the area of the trapezoid = 0.5*(4+1)*3 = 15/2

Finally, the required probability = trapezoid area / total area = (15/2)/12 = 12/24 = 5/8.

The correct answer is 5/8

Let X be a randomly selected real number from the interval [0,
1]. Let Y be a randomly selected real number from the interval [X,
1].
a) Find the joint density function for X and Y.
b) Find the marginal density for Y.
c) Does E(Y) exist? Explain without calculation. Then find
E(Y).

Let X represent the difference between the number of heads and
the number of tails when a coin is tossed 48 times. Then
P(X=8)=
So far I got 0.05946 but it keeps telling me I'm wrong

Let the random variable XX be the number of rooms in a randomly
chosen owner-occupied housing unit in a certain city. The
distribution for the units is given below.
XX
3
4
5
6
7
8
9
10
P(X)P(X)
0.10.1
0.270.27
0.350.35
0.160.16
0.040.04
0.040.04
0.020.02
?
(a) Is XX a discrete or continuous random variable? (Type:
DISCRETE or CONTINUOUS)
ANSWER:
(b) What must be the probability of choosing a unit with 10
rooms? P(X=10)P(X=10) =
(c) What is the...

Choose a random real number uniformly from the unit interval Ω =
[0, 1]. Consider the events: A = [1/2,1], B = [1/2,3/4], C =
[1/16,9/16]. Show that P(A ∩ B ∩ C) = P(A) · P(B) · P(C), but the
events A, B, C are not mutually independent.

Let X be a number chosen at random from the set
{1, 2, ... ,20} and let Y be a number chosen at random
from the set {1, 2, ... , X }. Let
pX |Y (x|y)
denote the condition distribution of X, given that
Y = y. Find
pX |Y (19|18)

Let X be a random number between 0 and 1 produced by a random
number generator. The random number generator will spread its
output uniformly (evenly) across the entire interval from 0 to 1.
All numbers have an equal probability of being selected. Find the
value of aa that makes the following probability statements
true.
(a) P(x≤a)=0.8
a=
(b) P(x < a) = 0.25
a=
(c) P(x≥a)=0.17
a=
(d) P(x>a)=0.73
a=
(e) P(0.15≤x≤a)=
a=

Given a random variable X following normal distribution with
mean of -3 and standard deviation of 4. Then random variable
Y=0.4X+5 is also normal.
(1)Find the distribution of Y, i.e. μy,σy
(2)Find the probabilities P(−4<X<0),P(−1<Y<0)
(3)Find the probabilities(let n size =8)
P(−4<X¯<0),P(3<Y¯<4)
(4)Find the 53th percentile of the distribution of X

If one three-digit number (0 cannot be a left digit) is
chosen at random from all those that can be made from the following
set of digits, find the probability that the one chosen is not a
multiple of 2. {0, 1, 2, 3, 4, 5, 6, 7}

A number x is selected at random in the interval [-3,3]. Let the
events A={x < 0}, B={|x - 0.5| < 0.5|}, and C={x > 0.75}.
Find P[A|B], P[B|C],P[A|CC],P[B|CC].

5. Pick a uniformly chosen random point inside the triangle with
vertices (0, 0), (3, 0) and (0, 3).
(a) What is the probability that the distance of this point to
the y-axis is less than 1?
(b) (b) What is the probability that the distance of this point
to the origin is more than 1?

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 8 minutes ago

asked 8 minutes ago

asked 8 minutes ago

asked 12 minutes ago

asked 18 minutes ago

asked 26 minutes ago

asked 38 minutes ago

asked 49 minutes ago

asked 59 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago