Statistics and Probability Answers

So I had this Probability Question in class, and I didn't understand how my teacher got that answer.

So the Question is:
Two fair dice is rolled 'n' times. Find the probability of obtaining a double six 'at least once'.

How I answered the question was that, if we want a Double six at least once (which means getting a Double six, any number of times, be it once, twice, thrice....n times)

So the complement of that would be not getting a double six at all
And the P(Not getting a double six at all) = 35/36

so the P(Getting a double six at least once) = 1 - P(Not getting a double six at all)
P(Getting a double six at least once) = 1 - 35/36 = 1/36

So the Probability of getting a Double Six at least once (n times) = (1/36)*(1/36)*(1/36)....n times
P(getting a Double Six at least once (n times)) = (1/36)^n

but my teacher said that the answer is = 1 - (35/36)^n

Now, I didn't understand the logic she applied so please please can you please solve this and tell me if my answer is right or explain how is my teacher's answer right! Please, this Questions has been eating my head!

If 500 people each select a number a random between 1 and 100, what is the probability
that 4 people select the number 25 ?

Heights of women have a bell shaped dist. mean of 160cm sd 0f 5cm using chebyshev's theorems, what are the minimum and maximum heights that are within 3 standard deviations of the mean

A medical doctor wishes to test the claim that the standard deviation of the systolic blood pressure of deep sea divers is greater than 450. To do so, she selected a random sample of 25 divers and found s = 468.

Assuming that the systolic blood pressures of deep sea divers are normally distributed, the doctor would perform a chi-square test to test her research hypothesis. In that case, what is the test value that she would compute.

The next 2 questions refer to the following scenario. Suppose the manager of a pet store wants to determine if there is a difference in the amount of money spent in the store, on average, by owners of dogs vs. owners of cats. Consider dog owners as group 1 and cat owners as group 2. Assume that the population standard deviations are equivalent between groups. Below are the sample data for the nine dog owners and the 9 cat owners. Test the hypothesis at alpha equals 1%.
The hypothesis at Ho: µ1 = µ2 vs. Ha: µ1 ≠ µ2.
Group1 Group 2
36 35.5
19 32.5
24.5 30
27 31.5
20 35.5
35 38
24.5 34.5
23.5 36
27.5 26

15) What is the p-value and test statistic used to test this hypothesis?
a) 0.0162 2.134
b) 0.9961 -2.134
c) 0.0339 3.044
d) 0.0756 3.044
e) 0.0077 -3.044

16) Is there a significant difference in the amount of money spent in the pet store, on average, by owners of dogs vs. c

Kindergarten children have heights that are approximately normally distributed about a mean of 39 in. And a standards deviation of 2 in. A random sample of size 25 is taken and the mean X is calculated. What is the probability that this mean value
Will be between 38.5 and 40.0 inches?

the one sample t statistic from a sample of n=42 observations for two sided test of for testing:
h0:µ = 81
h1:µ ≠ 81
has the value Z = 1.12. what is the P value of this test statistic?

During a typical one-hour lunch period 28 people go to a local deli. During a randomly chosen 15 minute period, what is the probability that:
a. Only 6 people go to the deli?
b. More than 12 people go to the deli?
c. At most 10 people go to the deli?
d. No more than 5 people go to the deli?

prepare a plan of CRD with 7 treatments and 35 plots available for them and copmlete first 2 columns of ANOVA

In a multi-user operating system, 20 requests are made to use a particular resource per hour on an average. The probability that no requests are made in 45 minutes is

a) e^(-15)
b) e^(-5)
c) 1-e^(-5)
d) 1-e^(-10)