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(a): there is a 8/10 probability of pulling a 3 and a 2/10 probability of pulling a 5 so the expected value is \(\frac{8}{10}(3) + \frac{2}{10}(5)\) Expected value for drawing one slip of paper, this is equal to 3.4 or 34/10 or 17/5

(b): well, this would be the exact same idea except the probabilities are different, now the expected value is \(\frac{9}{11}(3) + \frac{2}{11}(5)\) which simplifies to 37/11

(c): Once again, same idea, the expected value would be \(\frac{10}{12}(3) + \frac{2}{12}(5)\) which simplifies to 10/3

If you have any questions, just ask :)

There are two ways to get two black balls: either you draw two black balls on the first attempt, or you draw one black ball and one white ball on the first attempt, then draw two black balls on the second attempt.

The probability of drawing two black balls on the first attempt is:

P(draw two black balls on the first attempt) = (3/8) * (2/7) = 6/56 = 3/28

The probability of drawing one black ball and one white ball on the first attempt is:

P(draw one black ball and one white ball on the first attempt) = (3/8) * (5/7) + (5/8) * (3/7) = 30/56 = 15/28

If you draw one black ball and one white ball on the first attempt, then the probability of drawing two black balls on the second attempt is:

P(draw two black balls on the second attempt | one black ball and one white ball on the first attempt) = (3/8) * (2/7) = 6/56 = 3/28

To calculate the probability of drawing two black balls, we need to add the probability of the two cases:

P(draw two black balls) = P(draw two black balls on the first attempt) + P(draw two black balls on the second attempt | one black ball and one white ball on the first attempt)

P(draw two black balls) = (3/28) + (15/28)*(3/28) = 54/784 = 27/392

Sorry, thats incorrect! I appreciate your help tho :)

The answer is 3*sqrt(6).

It says thats wrong

Since the boxes are indistinguishable, we have the following 4 situations:

4 - 0 - 0, and there is only 1 option
3 - 1 - 0, and it is choose 1 out of 4, which has 4 options
2 - 2 - 0, and it is choose 2 out of 4, which has 6 options
2 - 1 - 1, and it is choose 2 out of 4, which has 6 options

The answer is then 1 + 4 + 6 + 6 = 17.

Can I have an answer please?

Your answer is useless.

I know, that this answer is a little late, so it may not help if this question was time sensitive, however, I hope this helps. For the product of the rolls to be one, we would need both rolls to be 1 which has a 1/6 * 1/6 = 1/36 probability of occuring, to have a product of 2, we would need a 1 and a 2 on the die rolls, which has a probability of 1/6 * 1/6 = 1/36 and for 3 we would need a 1 and a 3 which has a probability of 1/6 * 1/6 = 1/36. 1/36 + 1/36 + 1/36 = 3/36 = 1/12! If you have any questions, just ask :)

If this was a larger problem, I would have another method, however since this is a smaller variation of this question, you could just check all of the points on the coordinate grid that franklin could be once he takes his step. Also, I'm 90% sure that this is a repost of another problem, if you have any questions, just ask :)

This depends, do you have the side lengths for the triangle? If not, could you give me any lengths of the triangle that you do have? If you have any other questions, just ask :)

This question is similar to yours, I suggest looking at this link: https://web2.0calc.com/questions/in-how-many-ways-can-4-balls-be-placed-in-4-boxes, if you have any questions just ask :)

This is a repost of a previous question that has already been solved, here is the link: https://web2.0calc.com/questions/counting_37510, hope this helps :)

Well if you are round to only halfs and not whole numbers, then this would round down to 23 1/2, however if you are counting whole numbers, then this would round up to 24. If you have any more questions, just ask :)

Well if you wanted to simplify this, then you would distribute the 3 to the terms in the parentheses, 3(2x - 8) is equal to

6x - 24, 6x - 24 - 11x = -5x - 24. If you have any other questions, just ask. :)

We can begin by rearranging the given equation:

z - 3w - 2iw + 4iz = -8 + 12i

Simplifying the left side by grouping the real and imaginary terms of z and w separately:

z + 4iz - 3w - 2iw = -8 + 12i

Factor out z and w:

z(1+4i) - w(3+2i) = -8 + 12i

Now we can solve for one variable in terms of the other. Let's solve for z in terms of w:

z = (-8+12i+w(3+2i))/(1+4i)

To find all solutions for z and w, we need to substitute any possible values for w and solve for z.

Alternatively, we could solve for w in terms of z:

w = (-8+12i+z(1+4i))/(3+2i)

Again, we can substitute any possible values for z and solve for w.

Therefore, there are infinite solutions to this equation since we can choose any complex number for z or w, and the other variable can be calculated using the equation above.

The area of the largest square is 27^2 = 729.

The total area is then 729 + 729 (2/3) + 729 (2/3)^2 + 729 (2/3)^3 + 729 (2/3)^4 = 729 + 486 + 324 + 216 + 144 = 1899.

Thank you!

Thank you so much Alan!  I made a slight miscalculation when I tried multiplying by the conjugate.

This is an exact repost of a previous question, if you need help with this, I suggest going to this link: https://web2.0calc.com/questions/counting_93059.

Could you elaborate on what this route looks like, I'm guessing the total route has all flat, downhill, and uphill, but I have no total time to go on, thank you in advance.

If you understand how partitions work, I would probably try using partitions. If not, please reply to this post.

Well with only positive integers, what pairs of positive integers multiply to get 8, and what pairs of positive integers multiply to get 12. Then check which positive integers are similar between them. Hope you get the answer! (if not I'll be here most likely.)

I'm sorry, however your question is incomplete, please add the values of how many fractions there are and what the product equals. Thank you, and I will try to help once you have edited your question.