Mathematics
Grade 12
15 min
Write the probability distribution for a game of chance
Write the probability distribution for a game of chance
What you'll learn
- Read subtraction sentences with pictures and identify the total number of objects at the beginning of the problem with 80% accuracy.
- Use manipulatives (like counters) to act out subtraction problems and show the number being taken away with 80% accuracy.
- Write subtraction sentences (using numerals and the minus and equals signs) to match a given picture with 70% accuracy.
- Say the answer to a subtraction sentence aloud after solving with manipulatives with 90% accuracy.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Define a discrete random variable and its probability distribution.
Identify all possible outcomes and their associated net monetary values in a game of chance.
Calculate the probability of each outcome, expressing it as a rational number using principles of combinatorics.
Construct a complete probability distribution table for a given game of chance.
Calculate the expected value (mean) of a discrete random variable using its probability distribution.
Interpret the expected value to determine if a game is fair, favorable, or unfavorable to the player.
Ever wondered if that carnival game is *really* winnable? 🎡 Let's use the power of rational functions and probability to see if the odds are truly in your favor!
This tutorial will guide you through t...
2
Key Concepts & Vocabulary
TermDefinitionExample
Discrete Random Variable (X)A variable whose value is a numerical outcome of a random event and can only take on a finite or countably infinite number of distinct values.In a game where you flip 3 coins, the number of heads you get is a discrete random variable, as it can only be 0, 1, 2, or 3.
Probability DistributionA table or function that links every possible value of a discrete random variable with its corresponding probability of occurrence.For a fair six-sided die, the probability distribution would assign a probability of 1/6 to each of the values {1, 2, 3, 4, 5, 6}.
Net Outcome (x)The actual profit or loss associated with an outcome in a game of chance, calculated by subtracting the cost to play from the prize money.If a game costs $2 to play and you win a $...
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Core Formulas
Properties of a Probability Distribution
1. $0 \le P(X=x_i) \le 1$ for all outcomes $x_i$.
2. $\sum_{i=1}^{n} P(X=x_i) = 1$.
These two properties must always be true. First, the probability of any single outcome must be between 0 and 1 (inclusive). Second, the sum of the probabilities of all possible outcomes must equal exactly 1.
Probability as a Rational Function
P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{n(A)}{n(S)}
For events with equally likely outcomes, the probability is a ratio (a rational number). The numerator and denominator are often found using counting principles like combinations, $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
Expected Value Formula
E[X] = \sum_{i=1}^{n} x_i \cdot P(X=x_i) = x_1...
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Challenging
You pay $5 to draw one card from a standard deck. You win $50 if it's an Ace. You win $13 if it's any other face card (King, Queen, Jack). Otherwise, you lose. What is the expected value of this game?
A.$1.54
B.-$0.38
C.$0.00
D.-$2.15
Easy
In the context of a game of chance, which of the following best defines a 'discrete random variable'?
A.The cost to play the game.
B.variable representing a numerical outcome of a random event, such as net winnings.
C.The probability of winning the game.
D.The total number of possible outcomes in the game.
Easy
For any valid probability distribution for a discrete random variable X, which of the following properties must be true?
A.The sum of all possible values of X must be 1.
B.Each probability P(X=x) must be greater than or equal to 1.
C.The expected value E[X] must be equal to 1.
D.The sum of the probabilities for all possible outcomes must be equal to 1.
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Write the probability distribution for a game of chance is a Grade 12 Mathematics lesson on ExcelOS.
What will I learn in Write the probability distribution for a game of chance?
You'll be able to: Read subtraction sentences with pictures and identify the total number of objects at the beginning of the problem with 80% accuracy; Use manipulatives (like counters) to act out subtraction problems and show the number being….
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This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.