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STAT 20: Introduction to Probability and Statistics

- PS 7
- Concept Review
- Concept Questions
- Break
- PS 8 (computing probabilities)
- Break

- Multiplication rule

For two events \(A\) and \(B\), \(P(A \text{ and } B) = P(A \vert B) P(B)\)

- Complement rule

\(P(A^C) = 1 - P(A)\)

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Flip 3 coins, one at a time. Define the following events:

\(A\) is the event that the first coin flipped shows a head

\(B\) is the event that the first two coins flipped both show heads

\(C\) is the event that the last two coins flipped both show tails

The events A and B are: ________

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Flip 3 coins, one at a time. Define the following events:

\(A\) is the event that the first coin flipped shows a head

\(B\) is the event that the first two coins flipped both show heads

\(C\) is the event that the last two coins flipped both show tails

The events \(A\) and \(C\) are: ________

Suppose we draw 2 tickets at random without replacement from a box with tickets marked {1, 2, 3, . . . , 9}. Let A be the event that at least one of the tickets drawn is labeled with an even number, let B be the event that at least one of the tickets drawn is labeled with a prime number (recall that the number 1 is not regarded as a prime number). Suppose the numbers on the tickets drawn are **3** and **9**.

Which of the following events occur?

\(A\)

\(B\)

\(A\) and \(B\) (\(A \cap B\))

\(A\) and \(B^C\)

\(A^C\) and \(B\)

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The Houston Astros beat the Philadelphia Phillies in the 2022 World Series. The winners in the World Series have to win a majority of 7 games, so the first team to win 4 games wins the series (best of 7). The Astros were heavily favored to win, so the outcome wasn’t really a suprise. Suppose we assumed that the probability that the Astros would have beaten the Phillies in any single game was estimated at 60%, independently of all the other games, what was the probability that the Astros would have won in a clean sweep?

(Clean sweep means that they won in the first 4 games - which didn’t happen, they won in 6 games.)

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Suppose we assume, instead, that the probability that the Astros would have beaten the Phillies in any single game was 50%, independently of all the other games. In this case, was the probability that the series would have gone to 6 games higher than the probability that the series would have gone to 7 games, **given** that 5 games were played?

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Let’s play a game where I first roll a fair six-sided die, and then toss a coin as many times as the number of spots I rolled. I win the game if I get all heads.

Given I roll a k what is the probability that I flip all heads?

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Let’s play a game where I first roll a fair six-sided die, and then toss a coin as many times as the number of spots I rolled. I win the game if I get all heads.

What is the probability I win the game?

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A rare condition affects **0.2%** of the population. A test for this condition is 99% accurate: this means that the probability that a person *with* the condition tests positive is 99% and the probability that a person *without* the condition tests negative is 99%. What is the probability that a person who tests positive has the condition?

Let \(A\) and \(B\) be events with positive probability. Then:

- \(B\) can be written as \(B = (B\cap A) \cup (B \cap A^C)\)

(ii)\[ \begin{align} P(A|B) &= \displaystyle \frac{P(A \cap B)}{P(B)} \\ &= \frac{P(A \cap B)}{P(B\cap A) + P(B \cap A^C)} \\ &= \frac{P(B|A)P(A)}{P(B|A)P(A) + P(B|A^C)P(A^C)} \end{align}\]

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