Question

expected value probability worksheet - fall semester 2025 - room 404 - mr scheel

1. how many heads would you expect if you flip a coin:

a. 10 times
b. 25 times
c. 50 times
d. 1000 times
e. n times

1. how many 1s would you expect if you roll a 6 - sided die:

a. 12 times
b. 30 times
c. 100 times
d. 500 times
e. n times

1. how many non - 6s would you expect if you roll a 6 - sided die:

a. 24 times
b. 60 times
c. 200 times
d. 1000 times
e. n times

1. how many 10s would you expect if you roll a 20 - sided die:

a. 50 times
b. 100 times
c. 600 times
d. 2000 times
e. n times

1. how many multiples of 4 would you expect if you roll a 20 - sided die:

a. 20 times
b. 150 times
c. 500 times
d. n times

Explanation:

Step1: Determine the probability of the event

For a fair coin, the probability of getting heads \(P(H)=\frac{1}{2}\). For a fair \(n -\)sided die, the probability of a particular outcome \(k\) is \(P(k)=\frac{1}{n}\).

Question 1

• A. 10 times

The expected value formula is \(E(X)=n\times P(X)\). Here \(n = 10\) (number of trials) and \(P(\text{heads})=\frac{1}{2}\). So \(E(X)=10\times\frac{1}{2}=5\)

• B. 25 times

\(n = 25\), \(E(X)=25\times\frac{1}{2}=12.5\)

• C. 50 times

\(n = 50\), \(E(X)=50\times\frac{1}{2}=25\)

• D. 1000 times

\(n = 1000\), \(E(X)=1000\times\frac{1}{2}=500\)

• E. \(N\) times

\(E(X)=N\times\frac{1}{2}=\frac{N}{2}\)

Question 2

For a 6 - sided die, \(P(1)=\frac{1}{6}\)

• A. 12 times

\(n = 12\), \(E(X)=12\times\frac{1}{6}=2\)

• B. 30 times

\(n = 30\), \(E(X)=30\times\frac{1}{6}=5\)

• C. 100 times

\(n = 100\), \(E(X)=100\times\frac{1}{6}=\frac{50}{3}\approx16.67\)

• D. 500 times

\(n = 500\), \(E(X)=500\times\frac{1}{6}=\frac{250}{3}\approx83.33\)

• E. \(N\) times

\(E(X)=N\times\frac{1}{6}=\frac{N}{6}\)

Question 3

For a 6 - sided die, \(P(\text{non}-6)=\frac{5}{6}\)

• A. 24 times

\(n = 24\), \(E(X)=24\times\frac{5}{6}=20\)

• B. 60 times

\(n = 60\), \(E(X)=60\times\frac{5}{6}=50\)

• C. 200 times

\(n = 200\), \(E(X)=200\times\frac{5}{6}=\frac{500}{3}\approx166.67\)

• D. 1000 times

\(n = 1000\), \(E(X)=1000\times\frac{5}{6}=\frac{2500}{3}\approx833.33\)

• E. \(N\) times

\(E(X)=N\times\frac{5}{6}=\frac{5N}{6}\)

Question 4

For a 20 - sided die, \(P(10)=\frac{1}{20}\)

• A. 50 times

\(n = 50\), \(E(X)=50\times\frac{1}{20}=2.5\)

• B. 100 times

\(n = 100\), \(E(X)=100\times\frac{1}{20}=5\)

• C. 600 times

\(n = 600\), \(E(X)=600\times\frac{1}{20}=30\)

• D. 2000 times

\(n = 2000\), \(E(X)=2000\times\frac{1}{20}=100\)

• E. \(N\) times

\(E(X)=N\times\frac{1}{20}=\frac{N}{20}\)

Question 5

For a 20 - sided die, the multiples of 4 are \(4,8,12,16,20\). So there are 5 outcomes. \(P(\text{multiple of }4)=\frac{5}{20}=\frac{1}{4}\)

• A. 20 times

\(n = 20\), \(E(X)=20\times\frac{1}{4}=5\)

• B. 150 times

\(n = 150\), \(E(X)=150\times\frac{1}{4}=37.5\)

• C. 500 times

\(n = 500\), \(E(X)=500\times\frac{1}{4}=125\)

• D. \(N\) times

\(E(X)=N\times\frac{1}{4}=\frac{N}{4}\)

Answer:

1.
A. \(5\)
B. \(12.5\)
C. \(25\)
D. \(500\)
E. \(\frac{N}{2}\)
2.
A. \(2\)
B. \(5\)
C. \(\frac{50}{3}\approx16.67\)
D. \(\frac{250}{3}\approx83.33\)
E. \(\frac{N}{6}\)
3.
A. \(20\)
B. \(50\)
C. \(\frac{500}{3}\approx166.67\)
D. \(\frac{2500}{3}\approx833.33\)
E. \(\frac{5N}{6}\)
4.
A. \(2.5\)
B. \(5\)
C. \(30\)
D. \(100\)
E. \(\frac{N}{20}\)
5.
A. \(5\)
B. \(37.5\)
C. \(125\)
D. \(\frac{N}{4}\)