You, Ada, and Carlos are on a game show where the host will ask contestants to wager (or risk) money. Assume that you are all very smart, and that each of you will be willing to wager money if and only if there is a greater than 50% chance you will win that wager. The three of you each roll a 12-sided die (numbered 1 to 12) and keep your result private.
-The host asks Ada: “Would you like to wager that your number is higher than or equal to Carlos’s?”
-Ada says: “No”
-The host asks Carlos: “Would you like to wager that your number is higher than or equal to Ada’s?”
-Carlos says: “Yes”
-The host asks you: “Would you like to wager that your number is higher than or equal to Carlos’s?”
You rolled an 8.
a. Assuming that Carlos heard Ada’s conversation with the host, should you make the wager? If you did make it, what is the probability that you would win? Please explain your reasoning.
b. Now assume that Carlos did not hear Ada’s conversation before giving his response. Should you make the wager? What is the probability that you would win? Explain.
c. How certain do you need to be that Carlos heard Ada’s conversation for you to make the wager?
My thought process is as follows:
Since Ada declined the wager, her number must be either 1,2,3,4,5, or 6 since otherwise she would have a greater than 50% chance of being higher than another randomly rolled number(Carlos's number).
Carlos knows this and he also knows that the expected value of Ada's number is 3.5 so he says yes to the wager as long as his number is 4 or greater. So Carlos's number is either 4,5,6,7,8,9,10,11, or 12 which has expected value 8.
This means that should wager since we have a greater than 50% chance of winning since we win if Carlos has 4,5,6,7,8 which is a 5/9 chance.
Is this thinking correct?
Im still stuck on how to do part c. I am thinking that I do not wager because if Carlos chooses to wager without hearing Ada's information then he has either 7,8,9,10,11,12 which has expected value of 9.5 and we only have a 2/6 chance of winning if Carlos has a 7 or 8. But is there a way to get a specific probability for how certain we need to be that Carlos heard Ada's conversation?
Before you can attack part (c), you must analyze part (b), in the same way that you analyzed part (a).
$\underline{\text{Analysis of Part (b)}}$
Carlos' "yes" indicates that his number is some random element in $~\{7,8,9,10,11,12\}.~$ This implies that you have a $~\frac{1}{3}~$ probability of winning.
$\underline{\text{Analysis of Part (c)}}$
Let $~p~$ denote the probability that Carlos heard Asa's response. Then Part (c) reduces to determining the value of $~p~$ so as to render your wager a break-even proposition.
Then,
$$\frac{5}{9}p + \frac{1}{3}(1-p) = \frac{1}{2} \implies $$
$$\frac{1}{18} \left[ ~10p + 6 - 6p ~\right] = \frac{1}{18} \left[ ~9 ~\right] \implies $$
$$4p + 6 = 9 \implies p = \frac{3}{4}.$$