Let A and B be events with P(A)=0.6, P(B)=0.4, and P(B|A)= 0.4. Find P(A and B).

Let A and B be events with P(A)=0.6, P(B)=0.4, and P(B|A)= 0.4. Find P(A and B).

Question
Probability
asked 2020-10-27
Let A and B be events with P(A)=0.6, P(B)=0.4, and \(\displaystyle{P}{\left({B}{\mid}{A}\right)}={0.4}\). Find P(A and B).

Answers (1)

2020-10-28
From the definition of the conditional probability, \(\displaystyle{P}{\left({B}{\mid}{A}\right)}=\frac{{{P}{\left({A}\ {\quad\text{and}\quad}\ {B}\right)}}}{{P}}{\left({A}\right)}\)
So, \(\displaystyle{P}{\left({A}\ {\quad\text{and}\quad}\ {B}\right)}={P}{\left({A}\right)}{P}{\left({B}{\mid}{A}\right)}={0.6}\cdot{0.4}={0.24}\)
0

Relevant Questions

asked 2021-05-16
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
asked 2020-11-10
If two events A and B are independent and you know that P(A)=0.3, what is the value of P(A|B)?
asked 2021-06-06
Let X and Y be independent, continuous random variables with the same maginal probability density function, defined as
\(f_{X}(t)=f_{Y}(t)=\begin{cases}\frac{2}{t^{2}},\ t>2\\0,\ otherwise \end{cases}\)
(a)What is the joint probability density function f(x,y)?
(b)Find the probability density of W=XY. Hind: Determine the cdf of Z.
asked 2021-05-05
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with \(\displaystyle\mu={1.5}\) and \(\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}\).
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than \(\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}\).
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of \(\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}\)? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?
asked 2021-01-24
If you used a random number generator for the numbers from 1 through 20 to play a game, what is the theoretical probability of getting each of these outcomes? a. A multiple of 3 or a multiple of 7, P(multiple of 3 or multiple of 7) b. P( even or odd) c. P(prime or 1) d. How did you find the probabilities of these events?
asked 2021-05-12
Let \(X_{1},X_{2},...,X_{6}\) be an i.i.d. random sample where each \(X_{i}\) is a continuous random variable with probability density function
\(f(x)=e^{-(x-0)}, x>0\)
Find the probability density function for \(X_{6}\).
asked 2021-01-19
If P(A)=0.5, P(B)=0.4, and A and B are mutually exclusive, Find P(A or B). P(A or B)=____
asked 2021-03-23
Five distinct numbers are randomly distributed to players numbered 1 through 5. Whenever two players compare their numbers, the one with the higher one is declared the winner. Initially, players 1 and 2 compare their numbers; the winner then compares with player 3, and so on. Let X denoted the number of times player 1 is a winner. Find P{X = i}, i = 0,1,2,3,4.
asked 2021-05-02
Let X and Y be jointly continuous random variables wth joint PDF is given by:
\(f_{X,Y}(x,y)=2\) where \(0 \leq y \leq x \leq 1\)
Find \(P(X\geq \frac{1}{2})\).
asked 2020-11-08
According to Exercise 16, the probability that a U.S. resident has traveled to Canada is 0.18, to Mexico is 0.09, and to bith countries is 0.04.
What's the probability that someone who has traveled to Mexico has visited Canada too?
Are traveling to Mexico and to Canada disjoint events?
Are traveling to Mexico and to Canada independent events?
Explain.
...