Amy, Brett and cate each secretly write down z, u or t. What is the probability that's cate's letter is different from both any and bretts letter?

4/9

Step-by-step explanation:

Since they write the letter secretly, that means the event will be independent. Any person can write 1 letter out of 3 options, so there will be 3 different ways for each person.

Cate will always choose a letter, so there is only 1 possibility here. The chance for Brett/Amy to choose a different letter will be 2/3 since Cate chooses one of the 3 possible letters. The probability of both Amy and Bretts letter different from Cate will be: 1 * (2/3) * (2/3) = 4/9