Ana, Beto, and Carolina are going to participate in the television show "Discover the Color of Your Hat". In the show, they stand in a circle and a blue or green hat will be placed on each of their heads. Each can see the hats of the others, but not the color of their own hat. Then, each of them writes down one of three options: "blue", "green", or "pass". If all those who wrote "blue" or "green" correctly guess the color of their hat, they win a brand new car. If any of them guess the color of their hat as "blue" or "green" and get it wrong, all three lose. If all of them write "pass", then all three also lose. It is worth noting that they cannot use signals and cannot see the papers of the other participants. The three meet to try to come up with a strategy. Carolina starts, "none of us should write 'pass', we should guess between 'blue' and 'green', because if we all pass, we will lose." Beto reacts by saying, "I disagree, it's better if only Ana guesses the color of her hat, while I and Carolina write 'pass'. In this case, the chance of winning will be higher." Ana speaks up, "I had an idea, if we use my strategy, the probability of winning the car will be $\frac{3}{4}$."

(a) Following Carolina's idea, what is the probability of winning the car?

(b) Changing to Beto's idea, what is the probability of winning the car?

(c) Give an example of a possible strategy from Ana that makes the probability of winning the car be $\frac{3}{4}$.
Solution

(a) Following Carolina's idea, each person has two possible hat colors, in total there are $2 \cdot 2 \cdot 2=8$ possible guesses. Among them, there is only one favorable case: they all guess correctly. Therefore, the probability of winning the car is $\frac{1}{8}$.

(b) With Beto's idea, there are now only two possibilities: Ana's hat is either blue or green. With one guess, she has a probability of $\frac{1}{2}$ of guessing correctly and all three winning. Note that Beto and Carolina will not influence the outcome, as they will simply pass.

(c) Each person looks at the colors of the hats of their two companions. If they are of different colors, this person should pass. If they are of the same color, then this person should guess that their hat is of the other color. For example, if the hats of Ana, Beto, and Carolina are blue, blue, and green, respectively, then Ana and Beto should write "pass," as they see hats of different colors, while Carolina should write "green," which is the color different from the two hats she sees. In this case, all three would win the car. In total, there are 8 possibilities for the colors of the three hats. Note that, with Ana's strategy, the three lose only in two possibilities: all hats are green or all hats are blue. In the other six possibilities, there will be two hats of one color and one hat of the other, and, as in the example, all three win. We conclude that the probability of winning the car is $\frac{6}{8}=\frac{3}{4}$.