Consider the "cupcake experiment" we described in class, where students sequentially pull a cupcake from a hat and privately look at its color, and then publicly guess the color of the majority of cupcakes in the hat. Recall that the probability that the majority is pink is 1/2 and the probability that the majority was brown is also 1/2. In class, we assumed that all players gave an honest guess of the color of the majority given their available information. In this problem, we will explore a situation where some students are not completely honest. Imagine that the first two students really like the color pink and instead of always trying to guess the majority correctly, they will do one of the following: a. Blindly guess pink: with probability %, they will guess "pink" regardless of the color of the cupcake they pull or the guesses of other students. b. Guess honestly: with probability 1/4, they will actually try to guess the majority using the available information. This changes the probability that the first two students will guess a color given their signal (the cupcake they pulled). For example, if the majority is brown, the probability of guessing pink is: P(Blindly guess pink) + P(Guess honestly) x P(Pulling pink|brown majority) = X 0.833 We can show that whenever the first two students guess honestly, they will always guess the color of the cupcake they pulled. Now assume that the third student is impartial between the colors pink and brown and their only interest is guessing the correct color of the majority given all available information. Also, assume that the third student knows that students 1 and 2 were blindly guessing pink with probability % and honestly guessing with probability '4. Imagine that the first two students publicly announce that their guess is pink. What should the third student guess if they pull a brown cupcake? In other words, compute the probability that the majority of the cupcakes is pink given that the first two students guessed pink and the third student pulled a brown cupcake. Show all your work