VCE Methods Question Thread!

[abbytennant]

If someone could give me a hand with this probability question that'd be great

'Find the probability that, in three tosses of a coin, there are three heads, given that there is at least one head."

The wording has confused me and I was thinking 1/8 but the book got 1/7
Thanks

[poppingsoda]

How do you eliminate careless errors? Adding wrong, subbing incorrectly, etc.

[cosine]

If someone could give me a hand with this probability question that'd be great

'Find the probability that, in three tosses of a coin, there are three heads, given that there is at least one head."

The wording has confused me and I was thinking 1/8 but the book got 1/7
Thanks

Let H = heads
Let T = tails
Let 1H = At least one head



So to work out the numerator, we require the probability that we get THREE heads AND at least ONE head. You can use some logic here, so to get three heads in total, then we obviously need to get one of more heads; so in other words, this is just going to be the probability of getting three heads: 1/8 = 0.125

Now we need to work out the denominator, the probability of getting at least one head, so we have the following options:
HHH, HHT, HTH, HTT, THH, THT and TTH. Or, in other words, you can just quickly do 1-Pr(TTT) because there are no heads in all three tails = 1-0.125 = 0.875

So we have:

[poppingsoda]

Find the inverse of y = 2/(x+3)^2 -1, x E (-3,infinity)

I did the usual x and y swapping and found y = root(2/x+1) - 3 but when I graph it it seems to be incorrect (I checked by letting the calculator graph the inverse for me). Is there something I'm missing here?

[abbytennant]

Let H = heads
Let T = tails
Let 1H = At least one head



So to work out the numerator, we require the probability that we get THREE heads AND at least ONE head. You can use some logic here, so to get three heads in total, then we obviously need to get one of more heads; so in other words, this is just going to be the probability of getting three heads: 1/8 = 0.125

Now we need to work out the denominator, the probability of getting at least one head, so we have the following options:
HHH, HHT, HTH, HTT, THH, THT and TTH. Or, in other words, you can just quickly do 1-Pr(TTT) because there are no heads in all three tails = 1-0.125 = 0.875

So we have:

Thank you!! Much appreciated Very clear explanation.

[Stewart98]

Hey guys, I have a couple of questions which i worked out and got the wrong solution. Can someone show me where I went wrong on both questions?

question 1) Find all values of x between 0 and 2pi for which:    Sin x = -0.45
Therefore: x = sin^-1(-0.45)
                  x = -0.46675 which is negative in 3rd and 4th quadrant.
Angle 1 = -0.46675 - Pi
             = -3.60836
Angle 2 = 2Pi + 0.46675
             = 6.74995
----------------------------------------------------------------------------------------------------------------------------------
Question 2)Find all values of Theta, between 0 degrees and 360 degrees for which:    Cos Theta = -0.5
Therefore: Theta = Cos^-1(-0.5)
                            = 120 degrees
which is - in 2nd and 3rd quadrant.
Angle 1 = 180 - 120 
             = 60 degrees
Angle 2 = 120 - 180
             = -60 degrees

Hence the Red answers are the incorrect ones.

[JellyBeanz]

Hey guys, I have a couple of questions which i worked out and got the wrong solution. Can someone show me where I went wrong on both questions?

question 1) Find all values of x between 0 and 2pi for which:    Sin x = -0.45
Therefore: x = sin^-1(-0.45)
                  x = -0.46675 which is negative in 3rd and 4th quadrant.
Angle 1 = -0.46675 - Pi
             = -3.60836
Angle 2 = 2Pi + 0.46675
             = [blue=red]6.74995[/color]
----------------------------------------------------------------------------------------------------------------------------------
Question 2)Find all values of Theta, between 0 degrees and 360 degrees for which:    Cos Theta = -0.5
Therefore: Theta = Cos^-1(-0.5)
                            = 120 degrees
which is - in 2nd and 3rd quadrant.
Angle 1 = 180 - 120 
             = 60 degrees
Angle 2 = 120 - 180
             = -60 degrees

Hence the Red answers are the incorrect ones.

Angle 2 = 2Pi - 0.46675
             = 5.8164

You know that the angle is negative in the 4th quadrant, to find this angle you use 2Pi - (theta) not 2Pi + (theta), this is where you have gone wrong. Remember that the negative sign is just indicative of the quadrants that the angle is in, you do not take into consideration the (-) sign when finding the angles in the quadrant. I believe you have made the mistake of 2Pi - (-0.46675),

Angle 2 = 180 + 60
             = 240 degrees

Same mistake over here.

Just remember, that the negative sign is just indicative of the quadrants that they are in and are not used when finding the angles in those quadrants, for ex 2Pi - 0.46675, not 2Pi - (-0.44675)

[abbytennant]

Having some trouble getting my head around this problem:

Tickets in a game of chance can be purchased for $2. Each ticket has a 30% chance of winning $2, a 10% chance of winning $20, and otherwise loses. How much might you expect to win or lost if you play the game 100 times?

I've managed to make a probability distribution table:
x        Pr(X=x)
$0       0.6
$2       0.3
$20     0.1

That's as far as I can get. Any help would be appreciated

[Blueyneilo]

You would do 100(E(X)), the expected value 100 times to see the amount of winnings that you would get, E(X)=0(0.6)+2(0.3)+20(0.1)=2.6. You expect to get $2.60 per game. Therefore you would expect to get $260 from winnings over 100 matches. It costs $2 per game so that would cost $200 to play 100 games. The overall gain over 100 games is then (winnings - entry fee)=260-200=$60

[Gromekk]

Dilations question from cambridge book

Spoiler

For each of the following, state a transformation which maps the graph of y=f(x) to the graph of y=f₁(x)



I thought it would be a dilation of 3 from the y axis but the answer is 1/3 from y.

Edit: dw just realised I was reading the question wrong, will try again

edit2: worked it out

[Adequace]

In general, how do we tell if we need to use a CAS to complete a question in an exercise? I had to graph 280(8t -0.5t^2 -1.4), the worked solution just stated to use a CAS.

Realistic question to be solved by hand??

[qazser]

In general, how do we tell if we need to use a CAS to complete a question in an exercise? I had to graph 280(8t -0.5t^2 -1.4), the worked solution just stated to use a CAS.

Realistic question to be solved by hand??

Yup. If it contains decimals, use a CAS

[Syndicate]

In general, how do we tell if we need to use a CAS to complete a question in an exercise? I had to graph 280(8t -0.5t^2 -1.4), the worked solution just stated to use a CAS.

Realistic question to be solved by hand??

You can by turning decimal values into fractions, then solve by completing the square.

Example:
f(t) = 280(8t-3/2t^2- 7/5)

[CarrymetoUni]

First post in atarnotes, lets see if I don't sound too stupid.
How does one factorise the following quadratic expression?
6(x-3)^2+13(x-3)+5

[MightyBeh]

First post in atarnotes, lets see if I don't sound too stupid.
How does one factorise the following quadratic expression?
6(x-3)^2+13(x-3)+5

Given that (x-3) is a repeated factor there's probably an easier way but I just expanded it and factorised it from there. Working's in the spoiler if that didn't help.

Spoiler

All expanded, so collecting like terms is next.