Try thinking about the whole population of bottles.
X% are not defective.
Y% are defective.
Now you pick a bottle at random and drop it four times.
What can happen?
It could be defective and survive. (a%)
It could be not-defective and survive (b%)
It could be defective and break...
I think the answer is 1007 turns. You have to force the opponent to an end to catch him.
WLOG suppose you know he is at a location x < 1006. Then you can force him to the nearest side by simply playing x+1. He must go to x-1 to avoid capture, so you next play x, and so on. Eventually he is...
Let a = mk and b=nk, where n and m are integers, and k is some prime factor of a and b.
Claim: (a-b) is divisible by k.
(a-b) = (mk-nk) = k(m-n)
m-n is an integer because m and n are integers, hence (a-b) is divisible by k.
The reason we don't allow fractions here is just because when we are talking about prime numbers, we are only concerned with integers, because that's what makes a number "prime." I mean, if we allowed fractions, then every number would be composite, and in an infinite number of ways. That just...
Well, suppose p divides both P and q.
Then there is some integer m such that mp=P and some integer n such that np = q, right?
But now consider the difference between P and q, P - q. By the above we know that this is also equal to mp - np, or (m-n)p. Since m and n are integers, their...
So you have, after vela's last comment:
A2 - B2 = 502 - 402
Using Eq1, we get:
A2 - (L-A)2 = 502 - 402
A2 - (L2 - 2AL + A2) = 502 - 402
Can you go from here?
Here's what happens. The probability that Y takes on any particular value is actually zero, because the probability that X takes on any particular value is zero. What we can talk about is the probability that Y takes on a value in an interval.
So, for example, what is the probability that Y...
What's the unit axis? Does this mean "pick a point uniformly at random on [0,1]" ?
I'm not sure exactly what \theta you are using here, but it is true that doing something with \sin \theta for a properly defined \theta could lead you to the right answer.
But maybe it's easier to just work...
It seems that what you want is:
\sum^n_{k=1} f_{2k-1} = f_{2n}
So you showed this for n = 3.
Now you need to show that this statement being true for n=j IMPLIES that it is true for n=j+1. So assume that it's true for j, and show that then it is true for j+1.
Yes, draw some triangles. You want to know what the length of SX is, right? So make it the hypotenuse of a right triangle with vertices S,X, and the midpoint of OS. (Has to be a right triangle by symmetry). Then figure out what kind of special triangle it is and what its angles are, etc.
Well...I think you probably didn't copy your counts right, because your answer is right, but doesn't equal what you wrote above. (It's missing a 12).
But think of it this way. If you just threw two dice and took the average, it would be 7, right? So now 1/6 of the time, you get twice as much...
You don't need to graph anything.
Let t be the number of hours since noon that have elapsed.
Can you write down an expression in terms of t for how far apart the trains are?
(hint: they start 450 miles apart and one train leaves earlier)
Once you have done that, they will pass each other...
Does this pass the smell test? You are indicating that the standard deviation of the mean of the ten students is like 3.1 hrs.
But suppose we just took ten arts students (who have the higher standard deviation). You should know that the deviation of the group will be \frac{5}{\sqrt{10}}...
Well, no. That's the probability density function for the interarrival wait times if the process is Poisson. A particular wait time is always greater than zero, right?
It might be helpful if you gave us some context for this problem (ie, more of the problem if there is any, what class it's for).
To do this problem, we need more information. Is it a Poisson process? That's what I first thought of when I read the problem, but you don't mention that anywhere. Then you said that the probability of it occurring in any particular second was 1/2.6, which further makes it sound like a Poisson...
Do you mean the first element of the matrix has to equal one?
In any event, you can read more about Gaussian elimination here:
http://en.wikipedia.org/wiki/Gaussian_elimination
Basically, though, you can take a row in an augmented matrix and do a few different things to it.
1)...
Well, the cool thing is this obviously generalizes to any number of coins (n vs n+1); there's nothing special about 4 vs 3. If you multiply and sum out all the probabilities, you might be left with the idea that 4 vs 3 is like a special case that happens to end up as 1/2.
No, Alvin only wins when he has more heads. When they tie Andy wins.
(The original problem asked "what is the probability that Alvin gets more heads than Andy," so when we turn it into a game, Alvin wins if he gets more heads.)
A shortcut to this problem is this:
Compare Alvin's first 3 coinflips to Andy's first three coinflips. Sometimes they will be tied. If this occurs, then Alvin flips his 4th coin and wins (if its heads) or loses (if its tails). So he wins half of those. The rest of the time, someone will be...
On part 2, if a bet pays "X to 1" that means that if you win, you get your original bet back, plus X times your original bet. Gambling tables sometimes have a different designation "Y for 1" which means that your bet gets taken either way, and you get Y if you win.
So "5 to 1" and "6 for 1" are...
If you pick 35 random people out of the population, the size of your sample is 35.
If you pick one random person out of the population, the size of your sample is... one.
What is P(X - 50.5 > 28.87)? You might start by figuring out in words just what this expression means. It's pretty simple to get the answer once you understand the expression.