When doing this homework, you may ﬁnd the following identities useful: (see attached photo)
Albert has saved his homework in a ﬂash drive and put the drive in a drawer, without realizing that there are already 9 drives in that drawer. Now, there are 10 drives, and they all look identical.
If he now picks up the drives randomly one by one and checks whether it contains his homework, what is the probability that the kth drive that he picks up contains his homework, for k = 1,2, 3, . . . , 10? Summarize your result in a table to show the probability distribution.
Let X be the number of drives that he picks up until he ﬁnds his homework. Find the expected value E(X).
Find the variance V (X).
Every day Albert jogs on a circular track that is 5-kilometer long. Yesterday, he dropped a ﬂash drive somewhere on the track. He is going back today to look for the drive.
If he picks a random location on the track to start, and then walks in a clockwise direction until he ﬁnds the drive, what is the probability that the he will ﬁnd the drive after walking for no more than x kilometer?
Let X be the distance, measured in km, that he will have walked when he ﬁnds the drive. Find the expected value E(X).
Find the variance V (X).
Becky is playing a board game. She throws two fair, 6-faced dice with the numbers 1–6 on them, and the sum will determine how many spaces she will move.
Becky is playing a board game in which she glides along a path continuously instead of jumping from one space to another. She uses a random number generator to get two numbers from the continuous uniform distribution on [1,6], and the sum will determine how far she will move.
Hint: The uniform distribution is on [1,6], not [0,6]. Think about the implications.
If we pick a number k randomly from the uniform distribution on [0, 1], what is the probability that the equation x2 + 5kx + 1 = 0 has 2 distinct real roots?
Questions 6–10 are on the next page.
If we pick a number k randomly from the normal distribution with µ = 1, σ = 2; what is the probability that the equation x2 + kx + 1 = 0 has 2 distinct real roots?
what is the probability that this amount exceeds $110?
what is the probability that this amount is less than $105?
what is the probability that this amount is between $85 and $100?
There are 10 cards with the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 on them. If we randomly draw a card, with each card being equally likely to be drawn, what are the expected value and the variance?
In a standard deck of playing cards, there are 4 suits, and in each suit, there are 13 cards: A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, and K.
Suppose we count A as 1; and J, Q, and K as 10; and all the other cards at the numbers being shown. If we randomly draw cards from such a deck, with each card being equally likely to be drawn, what are the expected value and the variance?
Suppose we count A as 11; and J, Q, and K as 10; and all the other cards at the numbers being shown. If we randomly draw cards from such a deck, with each card being equally likely to be drawn, what are the expected value and the variance?
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