adijo/data-science-prep

Problems from https://datascienceprep.com/

data-science-prep's Issues

Wrong result for Q15:"Max Dice Roll"

Hi @adijo, I believe that in your PDF, the solution for question 15 "Max Dice Roll" is not the expected answer.

You write the formula for P(max{X1, ..., Xn} <= r) but we don't want the probability that the largest number rolled is less than (or equal to) r. We want the probability that the largest number rolled is equal to r.

So the expected result is P(max{X1, ..., Xn} = r) which changes the formula:

P(max{X1, ..., Xn} = r) = r^n/6^n - (r-1)^n/6^n
(from here)

You can verify that the PDF formula does not give the wanted result:

• for n=1 and r=2: "What is the probability that I roll a 2?" We know it's 1/6 but the formula gives 2/6.
• for r=6: "What is the probability that the largest number rolled is 6?". The formula gives a probability of 1, which means that whatever the number of rolls, we will get a 6! That would be nice!

Q21 in pdf

The total number of possibilities was considered as 8-1 =7

But why will the order matter here, isn't 101, 110 and 011 the same ?

I think there are three possibilities that contain at least one fan viz (fan, not fan, not fan), (fan, fan, not fan) and (fan, fan, fan)

out of these only one combination has exactly one fan, so probability is 1/3

Expected value of dice roll is 3.5, not 3 (question 6)

In Question 6 – Picking between two dice games, you state the expected value E[X] of a dice roll (X ∼ Uniform(1, 6)) is 3, when it is actually 3.5. This results in a very different expected value for E[XY] of 3.5 * 3.5 = 12.25 and E[Z^2] = 2.92 + 12.25 = 15.17.

Other than that, I enjoyed reading through your solutions. Cheers!

Question 10 - Flipping Game

I had the initial idea that P(HH) = P(TH) as well, but list out a few possibilities and you'll see that this isn't the case! In fact, this game is very biased towards one side. Try to consider what happens when the first coin flip yields T.