Permutations and coin flip probability questions

Question

Q:  An elf, a dwarf, a hobbit, a wizard, and a man are sitting in a line. How many different ways can they sit if the elf and the dwarf insist on sitting next to each other?

For this one, I think I need to use permutations, but I am not sure how to set it up, and solve.

Q: If a coin is flipped five times, the probability of getting heads all five times is ___________.

Now this one is a question I am not very good at (coin probability). The question wants you to then check all that apply to the answer (to fill in the blank):

• 1/2

• The same as getting tails all five times

• The same as rolling a prime number on a number cube five times

• 1/32

• The probability of getting this answer correct

• The same as rolling a 2 on a number cube five times

• Never

Thank you for any help!

Answer 1

Treat the dwarf and elf as one spot

you 4 for first spot, 3 for the second spot

and 2 for the third spot one choice for the last spot

so you get 4 • 3 • 2 • 1 = 24 ways.

but the dwarf and elf can be switched so you multiply by 2

24 • 2 = 48 different lines

coin flip probability of head is 1/2 each time

1/2 • 1/2 • 1/2 • 1/2 • 1/2 = 1/32 which is the same as all tails

Answer 2

here are the possibilities

e=elf, d=dwarf, h=hobbit, w=wizard, m=man

e d _ _ _ 3 choices for place 3, 2 choices for place 4, 1 choice for place 5

3*2*1=6 choices

_ e d _ _ 3 choices for place 1, 2 choices for place 4, 1 choice for place 5

3*2*1=6 choices

_ _ e d _ 6 choices again by the same reasoning

_ _ _ e d 6 choices again by the same reasoning

so far there are 4*6=24 choices

now you can have the following...

d e _ _ _ 6 choices

_ d e _ _ 6 choices

_ _ d e _ 6 choices

_ _ _ d e 6 choices

4*6=24 choices

24+24=48 choices

the probability of getting heads is 1/2

the probability of getting heads all five times is 1/2 * 1/2 * 1/2 * 1/2 * 1/2=1/32 is one answer

the same as getting tails all five times is another answer (probability of getting tails is 1/2)

the same as rolling a prime number on a number cube five times is another answer because on a number cube there are 6 digits and 3 of them are prime (2,3 and 5 are prime and 3/6=1/2)

Answer 3

Stratton

Number the positions in the line 1,2,3,4,5

How many ways can we place the elf and the dwarf if they are next to each other

1 2 3 4 5

E D

D E

E D

D E

E D

D E

E D

D E

So 8 ways.

For each of the 8 ways we can place the other three in 3x2x1 = 6 ways. (1st empty space choice of 2, 2nd empty space choice of 2, 3rd empty space no choice).

So we can place the creatures in 8 x 6 = 48 ways

For the coin. If it's a fair coin p (head on 1 toss) = p (tail on 1 toss) = 1/2

So p (HHHHH) = p(H) . p(H) . p(H) . p (H) . p (H) = 1/32

so 7 possible answers a b c d e f g. a b d g you should be able to work out if correct now.

c - number cube has 3 prime numbers, so chance of 1 is 3/6 = 1/2. you should be able to work out if c is correct now.

f - should have enough background now to decide this

e - hard to say. If you guess randomly at all these answers would be (1/2 ^ 7, I'm not sure what the answer is here.

Mike