Matias Aguirre
2022-07-19
Answered

Prove or disprove via proof that events X and Y can be independent and mutually exclusive if both of their probabilities are greater than 0.

You can still ask an expert for help

Makenna Lin

Answered 2022-07-20
Author has **16** answers

Two events $X$ and $Y$ are independent if and only if $P(X\cap Y)=P(X)P(Y)$. They are mutually exclusive if and only if $P(X\cap Y)=0$.

In this case, $P(X)>0$ and $P(Y)>0$, which means that $P(X)P(Y)>0$. If the events are independent, then $P(X)P(Y)=P(X\cap Y)>0$, which would imply that they are not mutually exclusive!

In other words, the events cannot be both independent and mutually exclusive if both of their probabilities are greater than $0$.

In this case, $P(X)>0$ and $P(Y)>0$, which means that $P(X)P(Y)>0$. If the events are independent, then $P(X)P(Y)=P(X\cap Y)>0$, which would imply that they are not mutually exclusive!

In other words, the events cannot be both independent and mutually exclusive if both of their probabilities are greater than $0$.

asked 2021-09-12

Ali starts driving for office in the morning at some time uniformly chosen in the interval [8:30,9.00]. His favorite radio program comes on at 8:30 and may last anywhere between 5 to 15 minutes with equal probability. What is the probability that he catches at least a part of the program?

asked 2021-09-15

A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.

$n=6,p=0.3,x=4$

$P\left(4\right)=$

asked 2021-12-14

A batch of 400 LEDS contains 7 that are defective. This is known to be the long-term average for the production. The company sells boxes with 50 LEDS in each box.

(a) What is the probability there are 1 or more defective bulbs in a box?

(b) If three LEDS are selected at random from a box, what is the probability that the third one selected is defective given that the first one selected was not defective and the second one selected was defective?

(c) Boxes are returned if 3 or more are defective. What is the probably a box will be returned?

(a) What is the probability there are 1 or more defective bulbs in a box?

(b) If three LEDS are selected at random from a box, what is the probability that the third one selected is defective given that the first one selected was not defective and the second one selected was defective?

(c) Boxes are returned if 3 or more are defective. What is the probably a box will be returned?

asked 2021-09-20

a) Find binomial probability, $P(X=5)$ .

b) Find the value of binomial probability,$P(X=0)$ .

c) FInd the binomial probability,$P(X=9)$ .

b) Find the value of binomial probability,

c) FInd the binomial probability,

asked 2022-02-14

For binomial distribution X, with $n=7\text{}\text{and}\text{}p=0.6$ , what is $P\left(X>3\right)$ ?

asked 2021-12-12

Complete the table and construct the probability histogram for a binomial random variable x with $n=6\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}p=0.5$ .

$$\begin{array}{|cccccccc|}\hline x& 0& 1& 2& 3& 4& 5& 6\\ p(x)\\ \hline\end{array}$$

asked 2021-11-23

The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by

$f(x)=\{\begin{array}{ll}\frac{10}{{x}^{2}}& x>10\\ 0& x\le 10\end{array}$

(a) Find $\{X>20\}$. (b) What is the cumulative distribution function of X? (c) What is the probability that, of 6 such types of devices, at least 3 will function for at least 15 hours? What assumptions are you making?