Question

We roll a pair of dice. If the sum of the dice is 7, you pay me $26. If the sum is not 7, I pay you the number of dollars indicated by the sum of the dice. What is your expected value for the game?

Answer 1

we got to multiply the amount you win by the probability you win it and add it all up

Answer 2

wait so 26 * 7?

Answer 3

no it has nothing to do with \(26\times 7\)

Answer 4

ok im confused

Answer 5

yes, i see that

Answer 6

do you know the probability you roll a 7 on two dice?

Answer 7

if not, we can start there, because you are going to need this
you are going to need the probability for every possible roll of the dice actually
do you know them? "no" is a fine answer, we can find them all

Answer 8

i think so but im not positive. all i remember was there was a chart that listed the probabilities or at least the amount when rolling two die....

Answer 9

yeah you got that chart? we will need it

Answer 10

yes i can look it up

Answer 11

ok good
all the denominators we will leave as \(36\) because you are going to have to add

Answer 12

ok that makes sense.

Answer 13

btw thank you for being understanding, all the other tutors have made me feel so dumb and i honestly just need help cause ive taken this class like twice already and im beyond brain fried on the subject

Answer 14

also, you do understand that if you roll a 2 you get $2, if you roll a 3 you get $3, if you roll a 4 you get $4 etc, right?

Answer 15

yes

Answer 16

ok so our job it so multiply the amount you win by the probability you win it, and add up all the numbers

Answer 17

i happen to remember them
the probability you roll a 2 is \(\frac{1}{36}\)

so one term will be \(2\times \frac{1}{36}=\frac{2}{36}\) but lets just leave it as \(2\times \frac{1}{36}\)

Answer 18

the next one will be \(3\times \frac{2}{36}\) because the probability you roll a 3 is \(\frac{2}{36}\)
you following along?

Answer 19

yeah so then would you add:

1/36 + 2/36 +3/36 etc...

Answer 20

careful lets go slow

Answer 21

it is not what you wrote, it is this
\[2\times \frac{1}{36}+3\times \frac{2}{36}+4\times \frac{3}{36}+5\times \frac{4}{36}+6\times \frac{5}{36}\] there is more

Answer 22

oh!! ok

Answer 23

\[\overbrace{2}^{\text{ amount you win}}\times \overbrace{\frac{1}{36}}^{\text{probability you win it}}+3\times \frac{2}{36}+4\times \frac{3}{36}+5\times \frac{4}{36}+6\times \frac{5}{36}\]

Answer 24

all the way down the line
lets leave off the rolling 7 for a moment, you lose for that one

Answer 25

so we have more terms
\[8\times \frac{5}{36}+9\times \frac{4}{36}+10\times \frac{3}{36}+11\times \frac{2}{36}+12\times \frac{1}{36}\]

Answer 26

the denomnators are all 36 so we just need to add up in the numerator
\[\frac{2\times 1+3\times 2+4\times 3+5\times 4+6\times 5+8\times 5+9\times 4+10\times 3+11\times 2+12\times 1}{36}\]

Answer 27

i told you this was a pain
but not terribly hard just annoying
lets use a calculator

Answer 28

i get \(210\)
http://www.wolframalpha.com/input/?i=2%2B3*2%2B4*3%2B5*4%2B6*5%2B8*5%2B9*4%2B10*3%2B11*2%2B12 so \[\frac{210}{36}\] then we have to subtract the amount we lose times the probability we lose it

Answer 29

\[\frac{210}{36}-26\times \frac{6}{36}\]

Answer 30

\[\frac{210-156}{36}\] is our answer

Answer 31

Thank you!!!

Answer 32

yw

Answer 33

make sure to pass this time!

Answer 34

hahah no kidding!!