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MTH4120: Midterm 1 Practice 1

Problem 1. It is known that \( \left(\Omega,\mathcal F, \mathbb P\right) \) is a probability space and \( W \)a random variable on this probability space. Which of the following objects is a properly defined real number?
(A) \( \mathbb P\left(11\right) \)
(B) \( W+\mathbb P(11) \)
(C) \( \mathbb P\left(2W\geq 11\right) \)
(D) \( \{W\geq 11\} \)
(E) \( \{W-11\}+\mathbb P\left(11\right) \)

Problem 2. Which of the following formulas is true for every pair of events \( A \) and \( B \)?
(A) \( \mathbb P\left(A^c\cap B\right) =1-\mathbb P\left(A\cap B\right) \)
(B) \( \mathbb P\left(A^c\cap B\right) =1-\mathbb P\left(A\cap B^c\right) \)
(C) \( \mathbb P\left(A^c\cap B\right) =P(A)-\mathbb P\left(A\cap B^c\right) \)
(D) \( \mathbb P\left(A^c\cap B\right) =\mathbb P(B)-\mathbb P\left(A\cap B\right) \)
(E) \( \mathbb P\left(A^c\cap B\right) =\mathbb P(A)-\mathbb P\left(A\cap B\right) \)

Problem 3. A player participates in a game that involves rolling a fair die. The player rolls the die. If the number is equal to 6, then the player loses the game and has to pay \( 6 \) dollars. If the number is odd, then the player wins the game and receives a payment of \( 8 \) dollars. If the number is equal to 2 or 4, then the player wins \( 4 \) dollars and rolls the die again. If the number in the second roll is equal to 5 then the player gets another \( 3 \) dollars (in addition to \( 4 \) dollars already earned from the first roll).
Construct a probability space and a random variable for the income that the player makes in this game (the income could be a positive or a negative number). Determine the cumulative distribution function of the obtained random variable and draw its graph. Calculate the expected income that the player makes by playing this game.

Problem 4. A box contains \( 99 \) balls labeled with number \( 1 \); \( 53 \) balls labeled with \( 2 \); \( 37 \) balls labeled with \( 3 \); \( 3 \) balls labeled with \( 4 \); \( 53 \) balls labeled with \( 5 \); and \( 95 \) balls labeled with \( 6 \). The balls whose labels are odd numbers are green and the balls whose labels are even numbers are red. One ball is taken from the box. It is observed that the ball is of green color. What is the probability that its label is greater than \( 2 \)?

Problem 5. Assume that \( X \) is a positive random variable and that \( Y=\frac1X \) and \( Z=\frac1{X+4} \). It is known that \[ \mathbb E[X]=5.7,\quad \mathbb E[Y]=0.28,\quad \mbox{ and }\quad\mathbb E[Z]=0.12.\] Evaluate \( \mathbb E\left[XY+YZ+ZX\right] \).

Problem 6. A random point is chosen uniformly inside the triangle with vertices \( (4,1) \), \( (13,10) \), and \( (4,10) \). Compute the expectation of the distance from this point to the \( x \)-axis.

Problem 7. The sample space \( \Omega \) is defined as \begin{eqnarray*} \Omega&=& \Big\{ x, \{y,z, x\}, \{y, z\}, y \Big\} \end{eqnarray*} The collection of events \( \mathcal F \) is defined as \( \mathcal F=2^{\Omega} \). Which of the following objects are events?
Note: There could be more than one correct answer. In order to get the credit you must select every correct answer and you must not select any of the wrong answers.
(A) \( \{x, y\} \)
(B) \( \{\{x, y\}\} \)
(C) \( \{\{y, z\}\} \)
(D) \( x \)
(E) \( \{x\} \)

Problem 8. There are \( M \) green and \( N \) red apples in a basket. We take apples out randomly one by one until all the apples left in the basket are red. What is the probability that at the moment we stop the basket is empty?