Expected Value

Part of the Statistics for ML section of Coddy's Machine Learning journey. Lesson 16 of 64.

A raffle: 70% of tickets win nothing, 25% win 5 dollars, 5% win 20. What is one worth?

Three bars for the raffle prizes 0, 5 and 20 dollars, with chances 0.7, 0.25 and 0.05. The bars stand on a beam like a seesaw, balanced on a triangle at 2.25, the expected value

Weight each prize by its chance and add: the expected value, the chart's balance point:

prizes = np.array([0, 5, 20])
chances = np.array([0.7, 0.25, 0.05])
print((prizes * chances).sum())

This prints 2.25, the long-run average, though no ticket pays it. rng.choice with p= draws tickets by those chances:

rng = np.random.default_rng(1)
tickets = rng.choice(prizes, size=10000, p=chances)
print(tickets.mean())

This prints 2.298. At 3 dollars, a buyer loses 0.75 a ticket. A model's average error on new data is an expected value too.

Challenge

Medium

A fairground wheel pays the tokens in prizes, each zone with its chance in chances. The starter spun it 4000 times into spins; the program reads a spin's price.

Print the expected prize, then the simulated average, both to 2 decimals.
Print whether a player gains or loses per spin in the long run, and how much.

For another wheel and a price of 1.5:

Expected prize: 1.1 tokens
Simulated average: 1.08 tokens
A player loses 0.4 tokens a spin

Try it yourself

import numpy as np

price = float(input())

# The prize of each zone of the wheel, in tokens, and its chance
prizes = np.array([0, 1, 4, 10])
chances = np.array([0.5, 0.3, 0.15, 0.05])

# Don't change the lines below: 4000 simulated spins
rng = np.random.default_rng(9)
spins = rng.choice(prizes, size=4000, p=chances)

# Write your code below

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Statistics for ML

3Probability by Simulation

Probability as Frequency The Law of Large Numbers Sums of DiceExpected ValueConditional Probability Recap: Dice Stall Odds

6Relationships

Reading Scatter Plots Correlation Correlation Matrix Correlation Heatmaps Correlation Is Not Cause Recap: Ice Cream and Sun

9Feature Scaling

Why Scale Features Min-Max Scaling Standardization StandardScaler Recap: Climbing Partners

12Final Challenges

Train Delays Air Quality Wine Chemistry Clinic Survey

Practice on your own: Online Python compiler