I am just passing time playing with a deck of cards and thought of a probability question. It is a very straight forward problem, but for some reason I cannot figure out the solution.
If I have a deck of 52 cards, shuffled, and I turn over one card at a time, what is the probability that I will see 2 of the same number in a row? (i.e., flipping over a 9, followed by another 9).
Let me know what you all think! Thank you.
Question: When I explained Grandi's series to a friend, they stated that addition is commutative, which is true. They wanted to know why the commutative property is allowed to be violated for this series. I suggested that it has to do with the fact that it is a series, and not just any normal addition. Could anyone help me explain why we don't break any laws? Thank you.
Slope intercept form is the form y=mx+b.
This means that in each equation, we must solve for y in terms of x and a constant, b.
To solve for y, we need to have it alone on one side. In order to do that, we must move x to the other side. In all three equations, x is being added to y on the left side, so in order to move x over, we must subtract x from the left side. But whatever we do to the left side we must do to the right side. So after one step, you should have
x + 2y - x= -4 - x
x + y - x= 3 -x
x + 3y - x= -6 - x
which is the same as
2y= -4 - x
y= 3 -x
3y= -6 - x
Now, the second equation is in slope-intercept form, but the first and third are not. We must apply the same algebraic strategy that we did in the first step to get y alone. For example, in the first equation, y is being multiplied by 2. So in order to get rid of the two, we must divide the y by 2. But we must also divide the WHOLE right side by two, not just the -4. Once you do that for the first and third equation, you should end up with
y= -2 - x/2
y= 3 - x
y= -2 -x/3
Does that help?
Very Good! One quick addition to that would be to say that θ = π/2 ± 2πn, where n=0,1,2,.. This is the same as rotating on the unit circle and getting back to the same place each time. Therefore the true answer should be e^(-π/2 ± 2πn), which is still a real number!
There is another way of solving this problem that uses a different technique. Can anyone figure that one out? It simplifies things greatly.
I found this video on youtube of an MIT Physics lecture. If you jump ahead to around 45 minutes, the professor offers a math problem that I found interesting.
It took me a long time to work through it, but if you know the trick, it can be solved in less than 10 seconds.
i^i, where i = sqrt(-1)
the clue is that i=e^i(pi/2 plusminus 2*pi*n), n=0,1,2,...
Have fun with it, let me know what you think.