Solutions to Selected Exercises#

Directions: For each of the probability experiments below:

  1. Enumerate all the values of the random variable.

  2. Calculate a few of the individual probabilities involved, and create a table comparing \(x\) and \(p(x)\).

  3. Try to find a pattern. If you can’t (yet), add a few more entries to your table.

  4. Write a (closed-form) probability distribution function for the experiment, and verify the distribution function has the required properties.

  5. Identify the distribution as binomial, geometric, negative binomial, hypergeometric or none of the above.

Homework Problems

  1. A math teacher has 15 boys and 15 girls in his class. Because he’s sexist, he calls on girls 60% of the time. During a class period when he calls on 10 students, let \(X\) be the random variable indicating the number of girls called on before the first boy is asked a question.

  2. Draw 5 cards from a standard deck. Let \(X\) be the random variable indicating the number of hearts in the 5-card hand.

  3. When playing indoor soccer, Robbie has an amazing ability to miss on breakaway “open goal” scoring opportunities. Robbie misses 30% of the times he’s shooting on an open goal, and he has 8 open goal scoring opportunities in one notable game. Let \(X\) be the random variable indicating the number of scoring opportunities that “go begging” (e.g. Robbie misses).

  4. A quality assurance inspector randomly selects and test 5 light bulbs from each lot of 100. Suppose there are 10 defective light bulbs per lot (major assumption, but bear with me), and let \(X\) be the random variable indicating the number of defectives per draw.

  5. Draw 5 cards from a standard deck. Let \(X\) be the random variable indicating the number of Kings in the 5-card hand.

  1. A gambler playing American Roullette keeps betting on “Odd” until it pays off. American Roullette wheels have numbers 1 through 36, 0 and 00. Let \(X\) be the random variable denoting the number of spins until the wheel comes up Odd.

Solution. The chance of “Odd” occuring on any given spin is as follows:

\[\frac{9}{19}\]

Note that \(X=\{1,2,3,\cdots\}\). We can then calculate the following probabilities which are shown in the table below:

\(x\)

\(p(x)\)

1

\(p(x=1)=\frac{9}{19}\)

2

\(p(x=2)=\left(\frac{10}{19}\right)\left(\frac{9}{19}\right)\)

3

\(p(x=3)=\left(\frac{10}{19}\right)^2\left(\frac{9}{19}\right)\)

4

\(p(x=4)=\left(\frac{10}{19}\right)^3\left(\frac{9}{19}\right)\)

\(\vdots\)

\(\vdots\)

x

\(p(x)=\left(\frac{10}{19}\right)^{x-1}\left(\frac{9}{19}\right)\)

  1. The underdog baseball team in the World Series has a 40% chance of winning each game. Find a probability distribution function that gives the probability the underdog wins in 4, 5, 6 or 7 games, or not all.

Solution. The chance of a “Win” occuring in any given game of the series is \(0.4\). Note that 𝑋={4,5,6,7}. We can then calculate the following probabilities which are shown in the table below:

\(x\)

\(p(x)\)

4

\(p(x=4)=(0.4)^4\)

5

\(p(x=5)=\binom{4}{1}(0.6)(0.4)^4\)

6

\(p(x=6)=\binom{5}{2}(0.6)^2(0.4)^4\)

7

\(p(x=7)=\binom{6}{3}(0.6)^3(0.4)^4\)

\(\vdots\)

\(\vdots\)

x

\(p(x)=\binom{x-1}{x-4}(0.6)^{x-4}(0.4)^4\)

  1. Suppose a couple is trying to have a baby, and let \(X\) be the numbers of “attempts” during times which pregnancy is likely to happen. If the chances of success are 10% per attempt, write a probability density function for how many attempts are needed to get pregnant. How many attempts would be required to have AT LEAST an 80% chance of being pregnant?

  2. The favored baseball team in the World Series has a 60% chance of winning each game. Find a probability distribution function that gives the probability the favorites win in 4, 5, 6 or 7 games, or not all.

  3. In the sexist math teacher problem, Let \(X\) be the random variable indicating the number of girls called upon before the third boy is called upon.

  1. Billy Joe Bob Vernon McSweeney plays ball for the Nighthawks and has to sell 10 Cow Patty Bingo raffle tickets. He knows Momma McSweeney, Daddy McSweeney, Granny McSweeney, Aunt Marry Ellen Jane McSweeney and Grandpa McSweeney are guaranteed to buy a ticket. So that leaves 5 more to sell to students in his classes and random folks at Wal-Mart. Suppose Billy Joe Bob Vernon has a 10% chance of success each time he approaches someone about buying a ticket. Let \(X\) be the random variable indicating the number of trials required:

  • Write a function for the number of trials required before one ticket is purchased.

  • Write a function for the number of trials required before all five tickets are purchased.

  1. Suppose the underdog in a 5-game series has a \(\frac{1}{3}\) chance of winning each game. Find a probability distribution function for the underdog winning.

  2. Dr. Sinn, an Arsenal Gunners fan, hopes when he dies to have the funeral in London with former Gunners players acting as pall bearers. Why? So the team can let him down one final time. Yes, sad but true, Tuesday’s loss away to Borussia Dortmund was not atypical. The Gunners win lots of games but often stumble in the biggest games. Let’s suppose that, in each big game they play, the Gunners have a 35% chance of mucking it up (or, as the Brits say,”bottling it”). Let \(X\) be the random variable indicating the number of games played since the last muck up. Write a probability density function for number of big wins the Gunners will earn before mucking it up again.

  3. Shero is playing blackjack and sees her first card is an Ace. Given a standard deck of 52 cards with no other cards showing, let \(X\) be the random variable denoting the blackjack “count” of Shero’s two-card starting hand. Aces can count as either 1 or 11, and all face cards count as 10. All the number cards have their face value.

  4. Given the indoor soccer miscue scenario, let \(X\) be the random variable indicating the number of misses before Robbie scores 3 goals.