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Probability

Sample spaces, outcomes and events for scenarios like a swim-meet final, distinguishing an outcome from the event containing it before assigning probabilities.

A Cool Brisk Walk Through Discrete Mathematics, Probability

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18 exercises

Exercise 1

At a swim meet, the competitors in the 100-m freestyle are Ben, Chad, Grover, and Tim. These four swimmers make up our sample space $\Omega$ for the winner of this heat. Is Chad $\in \Omega$ ?

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Exercise 2

Is Tim an outcome?

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Exercise 3

Is Ben an event?

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Exercise 4

Is { Chad, Grover } an event?

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Exercise 5

Is { Ben } an event?

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Exercise 6

Suppose I told you that Pr({Ben})=.1, Pr({Chad})=.2, Pr({Grover})=.3, and Pr({Tim})=.3. Would you believe me?

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Exercise 7

Suppose I told you that Pr({Ben, Chad})=.3, and Pr({Ben, Tim})=.4, and Pr({Grover})=.4. Could you tell me the probability that Ben wins the heat?

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Exercise 8

And what's the probability that someone besides Chad wins?

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Exercise 9

Okay, so we have the probabilities of our four swimmers Ben, Chad, Grover, and Tim each winning the heat at .1, .2, .4, and .3, respectively. Now suppose Ben, Chad, and Grover are UMW athletes, Tim is from Marymount, Ben and Tim are juniors, and Chad and Grover are sophomores. We'll define $U$ ={Ben,Chad,Grover}, $M$ ={Tim}, $J$ ={Ben,Tim}, and $S$ ={Chad,Grover}. What's Pr( $U$ )?

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Exercise 10

What's Pr( $J$ )?

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Exercise 11

What's Pr( $\overline{U}$ )?

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Exercise 12

What's Pr( $J \cup S$ )?

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Exercise 13

What's Pr( $J \cap S$ )?

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Exercise 14

What's the probability of a UMW junior winning the heat?

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Exercise 15

What's the probability that the winner is from UMW or a junior (or both)?

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Exercise 16

What's Pr( $J|U$ )?

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Exercise 17

What's Pr( $\overline{U}|J$ )?

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Exercise 18

Suppose 75% of Twitter users vote, whereas only about half of people in general vote. Now say that about one out of every three people are on Twitter. If you see someone emerge from a voting booth, what's the probability they have a Twitter account?

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