NUMB3RS S2

NUMB3RS - Season 2 - "Mind Games" - Right or Wrong

Topic: The Binomial Theorem | Complexity: Algebra 2 / Pre-Calculus / Statistics

TI Activity PDF

Activity Introduction

In the NUMB3RS episode "Mind Games," a psychic shocks Charlie by incorrectly guessing the color (black or red) of 25 cards in a row. As Charlie's colleague Larry point outs, "the probability of getting them all wrong is the same as getting them all right." In this activity, we will apply the Binomial Theorem to find the probability of correctly guessing outcomes for a series of events.

Workbook ProgressStep 1 of 5

Step 1: Expanding Three Guesses

For each card guessed, the outcome is either right (RR) or wrong (WW). Since the order of guesses does not matter for the final tally, the overall expansion of guessing three cards is represented by: (R+W)3=R3+3R2W+3RW2+W3(R + W)^3 = R^3 + 3R^2W + 3RW^2 + W^3

Because the card is equally likely to be red or black, the probability of guessing correctly is R=0.5R = 0.5, and guessing incorrectly is W=0.5W = 0.5. Note that R+W=1R + W = 1.

Use this expansion to find the probability of guessing exactly 2 cards correctly and 1 card incorrectly.

Calculate the probability of guessing exactly 2 of the 3 cards correctly (3R2W3R^2W):

Hint

Substitute $R = 0.5$ and $W = 0.5$ into the term $3R^2W$.

Binomial Distribution Sandbox

Adjust trials and success probability to explore how the distribution curve shifts.

Number of trials (n):10
Success probability (p):0.50
Quick Presets:
Probability Bar Chart
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Theoretical
Hover bars to view decimals

Run Guessing Simulation

Run 1,000 empirical trials to observe how frequencies converge.