Answer each of the following questions. Show your work. (a) Two tennis players, A and B, are playing in a tournament; the first 3 sets is declared the winner of the match (Best-of-5 match. No tie Assume that A is stronger and wins each set with probability 0.6 a of each set is independent of other sets. What is the probability player A will win the match?

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**Transcription for Educational Website:**

### Probability and Statistics Problems

**Question (a):**

Two tennis players, A and B, are playing in a tournament; the first person to win 3 sets is declared the winner of the match (Best-of-5 match. No tied sets allowed). Assume that A is stronger and wins each set with probability 0.6, and the outcome of each set is independent of other sets.

**Problem:**
What is the probability player A will win the match?

**Question (b):**

Let A and B be events such that \( P(A) = 0.45 \), \( P(B) = 0.3 \), and \( P(A \cup B) = 0.6 \).

(i) Find \( P(A|B) \) and \( P(B|A) \).

(ii) Are A and B mutually exclusive? Circle your answer: **YES** / **NO**

(iii) Are A and B independent? Circle your answer: **YES** / **NO**

**Question (c):**

Consider two independent random variables X and Y where \( E(X) = 10 \) and \( \text{Var}(X) = 7 \); Y is a geometric random variable with \( p = 0.2 \). Let T be the random variable given by \( T = 2X - 3Y + 4 \). Then:

(i) \( E(X^2) = \underline{\hspace{5em}} \).

(ii) \( E(T) = \underline{\hspace{5em}} \).

(iii) \( \text{Var}(T) = \underline{\hspace{5em}} \).
Transcribed Image Text:**Transcription for Educational Website:** ### Probability and Statistics Problems **Question (a):** Two tennis players, A and B, are playing in a tournament; the first person to win 3 sets is declared the winner of the match (Best-of-5 match. No tied sets allowed). Assume that A is stronger and wins each set with probability 0.6, and the outcome of each set is independent of other sets. **Problem:** What is the probability player A will win the match? **Question (b):** Let A and B be events such that \( P(A) = 0.45 \), \( P(B) = 0.3 \), and \( P(A \cup B) = 0.6 \). (i) Find \( P(A|B) \) and \( P(B|A) \). (ii) Are A and B mutually exclusive? Circle your answer: **YES** / **NO** (iii) Are A and B independent? Circle your answer: **YES** / **NO** **Question (c):** Consider two independent random variables X and Y where \( E(X) = 10 \) and \( \text{Var}(X) = 7 \); Y is a geometric random variable with \( p = 0.2 \). Let T be the random variable given by \( T = 2X - 3Y + 4 \). Then: (i) \( E(X^2) = \underline{\hspace{5em}} \). (ii) \( E(T) = \underline{\hspace{5em}} \). (iii) \( \text{Var}(T) = \underline{\hspace{5em}} \).
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Step 1

Probability: 

The probabilistic situation for each strategy (course of action) may lead to a number of different possible outcomes. A listing of all the possible outcomes of a random variable with each outcome’s associated probability of occurrence is called a probability distribution. The numerical value of a random variable depends upon the outcome of an experiment and may be different for different trials of the same experiment. The set of all such values so obtained is called the range space of the random variable.

If the random variable x is continuous, then its probability density function must satisfy the following two conditions:

1. P(x)0 ; -<x< (non-negative condition)

2. -P(x) dx = 1 Total Probability is always 1

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