Question

Probability question:
https://fbcdn-sphotos-g-a.akamaihd.net/hphotos-ak-ash4/1011124_10201587850720874_687351010_n.jpg

Answer 1

Something like:
\[\Large \frac{1 - (4C0+4C1+4C2)}{2^x}\]Now idk what value of x should be there? And is the numerator correct?

Answer 2

I think the denominator should be \(2^4\) (the total number of possibilities with four coin-tosses)

Answer 3

Oh. I was wondering it to be 2^8.

Answer 4

But the numerator... it should be \[\large2^4 - (_4C_0 + _4C_1 + _4C_2)\]

Answer 5

I imagine you approached this by using 'complements' ?

Answer 6

So that goes...
\[\Large 1 - Pr(heads \ less \ than \ 3)\]

Answer 7

\[\Large 1 - \frac{4C0 + 4C1 + 4C2}{2^4}\]

Answer 8

lol etc :3

Answer 9

Oh. Are you sure about not including 1 in the numerator? :O

Answer 10

Is there any other way to solve it too?

Answer 11

Sure :)
Directly.

At least three heads means either 3 heads or 4 heads...
Probability of there being exactly three heads is...
\[\Large \frac{4C3}{2^4}\]

And the chances of there being exactly four heads is
\[\Large \frac{4C4}{2^4}\]



So, "at least three heads" means "exactly three heads or exactly four heads"
These are mutually exclusive (you can't have both exactly three heads and exactly four heads at the same time) so to get the probability of their union, you just add their probabilities...
\[\Large Pr(\text{at least three heads}) = \frac{4C3}{2^4}+\frac{4C4}{2^4}\]

Answer 12

It can also be shown (albeit not in any manner that I remember) that
\[\Large 4C0 + 4C1 + 4C2 + 4C3 + 4C4 = 2^4\]

Answer 13

Which makes sense, since the left side is basically adding up all possible ways to toss a coin four times, and it should be \(2^4\)

Answer 14

Binomial Theorem, methinks :3

Answer 15

I see. but tell me about this method too :D

\[\Large 1 - \frac{4C0 + 4C1 + 4C2}{2^4}\]

Answer 16

I think it should be:
\[\Large \frac{1 -(4C0 + 4C1 + 4C2)}{2^4}\]

Answer 17

if you toss a coin 4 times..2,2,2,2...there are 16 possible things you can get in total..
You want to find the number of ways you can make 3 out of 4 and 4 out of 4.
(4C3+4C4)/16 should be the answer

Answer 18

OH NO! WAIT! MY BAD!!

Answer 19

That method is called 'complementation' (I think?)

The probability of all things happening is 1.

So what you're basically doing is subtracting from 1 the probability of your desired event NOT HAPPENING.


The thing is, sometimes, it is easier to count the instances of something not happening as opposed to something happening. So to get the probability of an event, you can either directly solve its probability, or solve for the probability of that event NOT happening, and then subtract it from 1.

Answer 20

RIGHT! I got it now. Thanks. I have a couple more.

Answer 21

^_^

Answer 22

http://www.iba.edu.pk/News/past_entry_test/Past_Entry_Test_BS.html
Q46

Answer 23

Integrals? <gulp> not my cup of tea, but... I'm nothing if not adventurous :D

Answer 24

Q46 Q58 Q66 Q72 Q79 Q81 Q83

Answer 25

Ah. No problem. Try others^

Answer 26

Q58?
\[\Large |2x - 2y|+|2y - 2x|= \]

Answer 27

2 can be factored out in both addends...
\[\large |2(x-y)|+|2(y-x)|\]\[\large 2|x-y|+ 2|y-x|\]

Answer 28

But then again...
\[\Large |y-x|=|(-1)(x-y)|=|-1||x-y|=|x-y|\]

Answer 29

So, the expression becomes
\[\Large 2|x-y|+2|x-y|\]
or simply \[\Large 4|x-y|\]

Answer 30

Sorry. Was away.
But why it doesn't equal to zero?

Answer 31

Well, you can look at it this way:
Let u = x - y
means -u = y - x

the equation becomes...
\[\Large |2u| + |2(-u)|=|2u|+|-2u| = |2u|+|2u| = 4|u|\]

Answer 32

Oh dang!
Okay, next,

Answer 33

For 66 it should be C, right?

Answer 34

Oh. Yes.

Answer 35

wait, no. B.

Answer 36

Nope... we're talking about the possible values of y, not x.
The square root of 36 - x^2
may not take negative values :)

Answer 37

OHHHH! right. Q72.

Answer 38

72... well, the determinant is affected when you multiply through an entire row or column by a scalar....

Say, you have a matrix... if you multiply an entire column by 2, then you double the determinant...


In this case, the second column was doubled, and the third column was tripled...

So effectively, the determinant will be increased by a factor of 6.

Answer 39

I see. Next.

Answer 40

PS Calculators aren't allowed.
Q79

Answer 41

Oh don't worry :3

Answer 42

This can be solved directly...
\[\large \tan^{-1}(1) = \frac{\pi }{4}\]

Answer 43

\[\Large \sec\left(\frac{\pi }{4}\right)= \frac1{\cos\left(\frac{\pi}{4}\right)}= \sqrt2\]

Answer 44

I'm getting a sense of it.
last but not the least 83.

Answer 45

Let's see... the slopes are downward, so you can bet it's something that decreases...

Answer 46

That actually rules out A, C, and D, already, because those are all nondecreasing...

And obviously, this graph has intervals where the function is decreasing...

Answer 47

Nope. I don't get it.

Answer 48

what does [x] means?

Answer 49

Oh LOL
The floor function :D

It takes the greatest integer that is less than or equal to a number.

It's easier to see for positive values of x...

If x is an integer, than [x] = x

If x is not an integer, [x] just takes the integer part... like

[2.3] = 2
[9.00341] = 9
[8] = 8
[56.99999] = 56

Answer 50

So roughly, the graph of the function [x] is

Answer 51

Back.

I never studied this. This is crap. :P

Answer 52

So how is it C now?

Answer 53

It isn't :)
It's B.

Answer 54

Umm, how is it B

Answer 55

Okay, let's look at the interval from 0 to 1 first.

Answer 56

[x] looks like this: a constant line (at the x axis)

Answer 57

Crud... how to put this...
[x] - x
You're subtracting the actual number from its integer part.
Hence, their difference gets bigger and bigger (negatively) as x increases...

Since its integer part does not increase until x reaches the next integer...

Answer 58

Give me the example in figures, man!

Answer 59

I have to leave now. Have to break the fast.
Please write your working down. I'll be back in a hour or so. Maybe. Thanks for all the help.

Answer 60

Let's look at a few points from 0 to 1

0.1:
[0.1] - 0.1
= 0 - 0.1
= -0.1


0.2
[0.2] - 0.2
= 0 - 0.2
= -0.2

0.5
[0.5] - 0.5
= 0 - 0.5
= 0.5


0.75
[0.75] - 0.75
= 0 - 0.75
= -0.75

0.99
[0.99] - 0.99
= 0 - 0.99
= -0.99

But notice at x = 1:
[1] - 1
=1 - 1
=0 (back to zero, lol, as it is in the graph)