Name: _________________

AP Statistics Handout: Lesson 5.5

Topics: discrete random variables, expected frequency, expected value

Lesson 5.5 Guided Notes

Discrete Random Variables
Roulette: Let X = winnings from a $100 bet on black

18 Black Spaces
18 Red Spaces
X= +$100 -$100 1 Green Space

P(X)

In your own words, describe the difference between a regular variable and a random variable:

Expected Frequency

If I were to play roulette 1000 times, how many times would black be expected to win? Show your work.

Expected Value

If I were to play roulette 1000 times, what would my average winnings be per play? Show your work.

Material adapted from the Skew The Script curriculum (skewthescript.org)

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Expected Value: the _____________ value of a chance process, after repeating ___________________.

a) Use the space below to interpret the expected value of playing roulette (-$2.80):

b) Someone says, “If I bet $100, I’ll either win $100 or lose $100. So, how could the expected value
-$2.80? Losing $2.80 is not even a possible outcome of the game!” In your own words, answer
their question.

Calculating expected values

Example 1: You play a coin toss game. Heads means you get $100, tails means you lose $100. Find the
expected value.

Example 2: You play a coin toss game. Heads means you get $300, tails means you lose $100. Find the
expected value.

Example 3: You play a coin toss game. The coin is weighted so that there is a 75% chance of tails. Heads
means you get $100, tails means you lose $100. Find the expected value.

Example 4: You play a coin toss game. The coin is weighted so that there is a 75% chance of tails. Heads
means you get $300, tails means you lose $100. Find the expected value.

Material adapted from the Skew The Script curriculum (skewthescript.org)

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Expected Value for The Casino

Expected loss for 1,000 plays:

X= +$100 -$100

P(X) Expected casino profit for 1,000 plays:

Individuals don’t have many trials, but casinos do. They profit through ___________________ expected
values.

Lesson 5.5 Discussion

Imagine the casinos didn’t take AP Stats. They come up with this game for high-rollers: You roll a 6-sided
dice. If you roll 5-6, you get $1,000,000. If you roll 1-4, you have to pay $300,000.

Discussion Question: If you could play this game only

X= +$1,000,000 -$300,000 once, would you play? Why or why not? Mention the
expected value in your answer.

P(X)

Lesson 5.5 Practice

1) An art gallery owner has found that, of the people who enter the gallery, 8% tend to purchase at least
one piece of artwork. Over the course of a week, 130 people enter the gallery. How many people should
the owner expect to purchase at least one piece that week?

Material adapted from the Skew The Script curriculum (skewthescript.org)

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2) In a certain video game, there is a daily chest you can open to win free coins. Let 𝑋 = the number of
coins you win when opening the daily chest. The probability distribution of 𝑋 is shown below:

𝑋= 5 10 20 50

𝑃(𝑋) 0.75 0.15 0.07 0.03

a) If you open the chest 30 times, about how many times can you expect to win the 50 coin prize?

b) If you open the chest 100 times, about how many times can you expect to win at least 20 coins?

c) Find and interpret the expected value of 𝑋.

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3) A group of students were surveyed on how many siblings they have, and the answers varied from 0 to
4. Let 𝑆 = the number of siblings for a randomly selected student from this group. The probability
distribution for 𝑆 is shown below.

𝑆= 0 1 2 3 4

𝑃(𝑋) 0.3 0.4 0.1 0.15 0.05

If one student is selected at random, what is probability that the selected student has more than the
expected number of siblings?

4) A carnival game invites you to guess which cup is hiding a ball underneath it, where there are 4 cups
and one ball. In order to play, you must be blindfolded, so that you can’t see where the ball is initially
placed. You have to pay $5 to play, but if you win, they give you $20.

a) Complete the probability distribution below.

𝑋 = Player ' s net win

𝑃(𝑋) 0.75 0.25

Material adapted from the Skew The Script curriculum (skewthescript.org)

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b) Find and interpret the expected value.

c) Should the carnival expect this game to be profitable in the long run? Why or why not?

Further Practice
Teachers: If you’d like to give students additional practice problems…
• Check out our CED and Textbook Alignment Guide (skewthescript.org/ap-stats-alignment) to
find additional exercises in your AP Stats textbook or in AP Classroom that are aligned to the
content covered in this lesson.
• The following Free Response Questions (FRQs) from past AP Exams are also aligned to this
lesson: 2015 Q3, 2012 Q2, 2019 Q5 (part c)

Material adapted from the Skew The Script curriculum (skewthescript.org)