## anonymous 5 years ago Does anyone have any questions Something challenging

1. anonymous

solve for x (3 dp) a) 3^x=19. b) 2^1-x=7 c) (1/4)^x=75. How to do this?

2. anonymous

You need to use logarithms

3. anonymous

a) if 3 ^ x = 19 x = log 19 to the base 3 x = 2.68

4. anonymous

for b) 2^1 = 2 2 - x =7 -x = 7-2 x = -5

5. anonymous

for c) if (1/4) ^ x = 75 x = log 75 to the base 0.25 x = -3.11

6. anonymous

I think for b, I got 1.807.. and for a and c I got the same answer as yours.

7. anonymous

8. anonymous

the formula is if x^y = z y = log z to the base x

9. anonymous

hm.. ok.

10. anonymous

You understand logarithms right

11. anonymous

hm.. yup I understand it.

12. anonymous

So then you should show this as your method

13. anonymous

adam an eve take turns flipping a coin. the first one to flip 'heads' wins. if eve goes first, what it the probability she wins?

14. anonymous

1/2 ?

15. anonymous

nope. she has the advantage of going first, so it must be greater than 1/2.

16. anonymous

I dont think so The probability of her getting heads is half The probability of Adam getting heads and her cetting tails is 0.5 * 0.5 thats 1/4 So she should be 1 - 0.25 = 0.75

17. anonymous

no that is not it either but it is getting closer. two methods (which really amount to the same thing) a linear equation, or write out sample space of 'eve wins' and sum the geometric series.

18. anonymous

No i think it should just be 0.75 I can't think of anything else

19. anonymous

20. anonymous

how could she win? get heads on first try, done. i represent as (h) and probability is 1/2 eve gets tails, adam gets tails, and then eve gets heads. represent as (t t h) and probability is 1/8 eve gets tails, adam gets tails, eve gets tails, adam gets tails, eve gets heads (t t t t h) probability is 1/32 etc these events are mutually exclusive, so add them up $\frac{1}{2}+\frac{1}{8} +\frac{1}{32}+...$

21. anonymous

that is method one. method 2: put P = probability eve wins. write a linear equation in P as follows. she could win on first try with probability 1/2 OR she flips tails, adam flips tails [this happens with probability 1/4], AND she gets another turn. now if she gets another turn that is just like starting the game over, so her probability of winning has not changed. OR means we add the probabilities, AND means we multiply (assuming they are independent, which they are) and so we get an equation in P $P = \frac{1}{2} + \frac{1}{4}P$ If you sum the geometric series you see that it is identical so solving the linear equation for P