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Suppose that from a standard deck, you draw three cards without replacement. What is the expected number of spades that you will draw

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Answer 1

Answer:

Answer: The expected number of spades that you will draw is 0.751 spades

Step-by-step explanation:

The expected value can be calculated as:

∑xₙ*pₙ

Where xₙ is the n-th event, and pₙ is the probability of that event.

First, let's count the possible events and calculate the probability for each one.

x₀ = drawing 0 spades.

Out of 52 cards, we have only 13 spades, then 52 - 13 = 39 are not spades.

Then the probability of not drawing a spade in the first draw is:

p1 = 39/52

In the second draw we will have a card less than before in the deck (so we have 38 cards that are not spades, and 51 cards in total), then the probability of not drawing a spade is:

p2 = 38/51

And with the same reasoning, in the third draw the probability is:

p3 = 37/50

The joint probability for this event will be:

p₀ = p1*p2*p3 = (39/52)*(38/51)*(37/50) = 0.413

Second event:

x₁ = drawing one spade.

Let's suppose that in the first draw we get the spade, the probability will be:

p1 = 13/52

In the second draw, we get no spade, then the probability is:

p2 = 39/51

in the third draw we also get no spade, the probability is:

p3 = 38/50

And we also have the case where the spade is drawn in the second draw, and in the third draw, then we have 3 permutations, this means that the probability of drawing only one spade is:

p₁ = 3*p1*p2*p3 = 3*(13/52)(39/51)*(38/50) = 0.436

third event:

x₂ = drawing two spades:

Let's assume that in the first draw we do not get a spade, then the probabilities are:

p1 = 39/52

p2 = 13/51

p3 = 12/50

And same as before, we will have 3 permutations, because we could not draw a spade in the second draw, or in the third, then the probability for this case is:

p₂ = 3*p1*p2*p3 = 3*( 39/52)*(13/51)*(12/50) = 0.138

And the last event:

x₃ = drawing 3 spades.

The probabilities will be:

p1 = 13/52

p2 = 12/51

p3 = 11/50

And there are no permutations here, so the joint probability is:

p₃ = p1*p2*p3 = (13/52)*(12/51)*(11/50) = 0.013

Now we can calculate the expected value:

EV = 0*0.413 + 1*0.436 + 2*0.138 + 3*0.013 = 0.751

The expected number of spades that you will draw is 0.751 spades

Answer 2

Answer:

The expected number of spades drawn when drawing three cards without replacement from a standard deck is approximately 0.75 spades.

To calculate this, we can use the concept of conditional probability. Initially, there are 13 spades out of 52 cards in the deck, giving us a 13/52 chance of drawing a spade on the first card.

If the first card drawn is a spade, there are now 12 spades left out of 51 cards, so the probability of drawing a spade on the second card is 12/51.

If the first two cards are spades, there are 11 spades left out of 50 cards for the third draw, with a probability of 11/50.

Now, we multiply these probabilities together and sum up the possible scenarios (0, 1, 2, or 3 spades drawn) to get the expected value: (0 * (39/52 * 38/51 * 37/50)) + (1 * (13/52 * 39/51 * 38/50 + 39/52 * 12/51 * 38/50 + 39/52 * 38/51 * 11/50)) + (2 * (13/52 * 12/51 * 39/50 + 13/52 * 39/51 * 11/50 + 39/52 * 12/51 * 11/50)) + (3 * (13/52 * 12/51 * 11/50)) ≈ 0.75 spades.

So, the expected number of spades drawn when selecting three cards without replacement from a standard deck is approximately 0.75.

This means, on average, you can expect to draw about 3/4 of a spade.

for such more questions on number

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Related Questions

A bacterial culture starts with 500 bacteria and doubles in size every half-hour.(a) How many bacteria are there after 3 hours?Answer: Since 3 hours equals 6 half-hours, the culture will have doubled 6 times.Therefore, there will be500 · 26 = 32,000bacteria.(b) How many bacteria are there after t hours?Answer: Since t hours is the same as 2t half-hours, the culture will have doubled 2ttimes. Therefore, there will be500 · 22tbacteria.(c) How many bacteria are there after 40 minutes?Answer: There are two possible answers depending on how you interpret the set-upto the problem. If each bacterium in the culture doubles once every half-hour on thehalf-hour, then each one will double after exactly 30 minutes, and then not again until60 minutes have passed. In that case, there will be500 · 2 = 10004bacteria after 40 minutes.On the other hand, if each bacterium doubles exactly once per half-hour, but at somerandom time within that half-hour, then it makes sense to think of the populationfunction P(t) = 500 · 22t as continuous. In that case, since 40 minutes is4060=23of an hour, the population will be500 · 2223 = 500 · 243 ≈ 1259after 40 minutes.
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Which expression equals 24?
Which is the positive slope m, where m<1
Help me PLSSSSSSSSSSSSS

Find the equation of the axis of symmetry in the coordinates of the vertex of the graph of the function y=4x^2-8x-3

Answers

Answer:

the axis of symmetry is x=1

Good luck

Step-by-step explanation:

Answer:

x = 4; (4, –25)

Step-by-step explanation:

Confirmed in class

Ten years ago 53% of American families owned stocks or stock funds. Sample data collected by the Investment Company Institute indicate that the percentage is now 46% (the Wall Street Journal, October 5, 2012)a. Develop appropriate hypotheses such that rejection of H0 will support the conclusion that a smaller proportion of American families own stocks or stock funds in 2012 than 10 years ago.
b. Assume the Investment Company Institute sampled 300 American families to estimate that the percent owning stocks or stock funds was 46% in 2012. What is the p-value for your hypothesis test?
c. At α = .01, what is your conclusion?

Answers

Using the z-distribution, as we are working with a proportion, it is found that:

a) $H_0: p = 0.53$ , $H_1: p < 0.53$

b) The p-value is of 0.0075.

c) Since the p-value of the test is of 0.0075 < 0.01 for the left-tailed test, it is found that there is enough evidence to reject the null hypothesis and conclude that a smaller proportion of American families own stocks or stock funds in 2012 than 10 years ago.

What are the hypothesis tested?

At the null hypothesis, it is tested if the proportion is still of 53%, that is:

$H_0: p = 0.53$

At the alternative hypothesis, it is tested if the proportion is now smaller, that is:

$H_1: p < 0.53$

Item a:

The hypothesis are:

$H_0: p = 0.53$

$H_1: p < 0.53$

Item b:

The test statistic is given by:

$z = \frac{\overline{p} - p}{\sqrt{(p(1-p))/(n)}}$

In which:

$\overline{p}$ is the sample proportion.

p is the proportion tested at the null hypothesis.

n is the sample size.

In this problem, the parameters are:

$\overline{p} = 0.46, p = 0.53, n = 300$ .

Hence, the value of the test statistic is given by:

$z = \frac{\overline{p} - p}{\sqrt{(p(1-p))/(n)}}$

$z = \frac{0.46 - 0.53}{\sqrt{(0.53(0.47))/(300)}}$

$z = -2.43$

Using a z-distribution calculator, considering a left-tailed test, as we are testing if the proportion is less than a value, with z = -2.43, it is found that the p-value is of 0.0075.

Item c:

Since the p-value of the test is of 0.0075 < 0.01 for the left-tailed test, it is found that there is enough evidence to reject the null hypothesis and conclude that a smaller proportion of American families own stocks or stock funds in 2012 than 10 years ago.

More can be learned about the z-distribution at brainly.com/question/26454209

Answer:

a) Null hypothesis: $p\geq 0.53$

Alternative hypothesis: $p < 0.53$

b) $z=\frac{0.46 -0.53}{\sqrt{(0.53(1-0.53))/(300)}}=-2.429$

$p_v =P(Z<-2.429)=0.0076$

c) So the p value obtained was a very low value and using the significance level given $\alpha=0.01$ we have $p_v<\alpha$ so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 1% of significance the proportion of American families owning stocks or stock funds is significantly less than 0.53 .

Step-by-step explanation:

Data given and notation

n=300 represent the random sample taken

$\hat p=0.46$ estimated proportion of American families owning stocks or stock funds

$p_o=0.53$ is the value that we want to test

$\alpha=0.01$ represent the significance level

Confidence=99% or 0.99

z would represent the statistic (variable of interest)

$p_v$ represent the p value (variable of interest)

Concepts and formulas to use

Part a

We need to conduct a hypothesis in order to test the claim that proportion is less than 0.53 or 53%.:

Null hypothesis: $p\geq 0.53$

Alternative hypothesis: $p < 0.53$

Part b

When we conduct a proportion test we need to use the z statistic, and the is given by:

$z=\frac{\hat p -p_o}{\sqrt{(p_o (1-p_o))/(n)}}$ (1)

The One-Sample Proportion Test is used to assess whether a population proportion $\hat p$ is significantly different from a hypothesized value $p_o$ .

Calculate the statistic

Since we have all the info requires we can replace in formula (1) like this:

$z=\frac{0.46 -0.53}{\sqrt{(0.53(1-0.53))/(300)}}=-2.429$

Statistical decision

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.

The significance level provided $\alpha=0.01$ . The next step would be calculate the p value for this test.

Since is a left tailed test the p value would be:

$p_v =P(Z<-2.429)=0.0076$

Part c

So the p value obtained was a very low value and using the significance level given $\alpha=0.01$ we have $p_v<\alpha$ so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 1% of significance the proportion of American families owning stocks or stock funds is significantly less than 0.53 .

Find the volume of the cone below.

Answers

Answer:

$V =(1)/(3) \pi r^2 h$

For this case we know that the radius is r=7cm and the height is h =11cm. And replacing we got:

$V=(1)/(3) \pi (7cm)^2 (11cm)= (1)/(3) \pi (49cm^2) (11 cm)=(539)/(3) \pi cm^3$

And the best option would be:

$V = (539)/(3) \pi cm^3$

Step-by-step explanation:

For this case we know that the volume of the cone is given by:

$V =(1)/(3) \pi r^2 h$

For this case we know that the radius is r=7cm and the height is h =11cm. And replacing we got:

$V=(1)/(3) \pi (7cm)^2 (11cm)= (1)/(3) \pi (49cm^2) (11 cm)=(539)/(3) \pi cm^3$

And the best option would be:

$V = (539)/(3) \pi cm^3$

I'm having trouble with finding the solutions.

Answers

Answer:x=−8 and y=8

Step-by-step explanation: Do you want it solved step by step?

Answer:

One solution, (-8,8)

x = -8

y = 8

Step-by-step explanation:

-6x + 3y = 72 then 3y = 72 + 6x, y = 24 + 2x

5x + 8y = 24

substitute for y:

5x + 8(24 + 2x) = 24

5x + 192 + 16x = 24

21x = -168

x = -8

y = 24 + 2(-8) = 8

The sides of a square field are 24 meters. A sprinkler in the center of the field sprays a circular area with a diameter that corresponds to a side of the field. How much of the field is not reached by the sprinkler? Round your answer to the nearest hundredth. Use 3.14 for π.

Answers

Answer:

Step-by-step explanation:

i took the test

Factor out the GCF of (2x+1)