12th Grade
How do You Find the Expected Payoff in a Game of Chance?
{
"voice_prompt": "Speak clearly and naturally. Pause briefly at commas.",
"manuscript": {
"title": {
"text": "How do You Find the Expected Payoff in a Game of Chance?",
"audio": "How do You Find the Expected Payoff in a Game of Chance?"
},
"description": {
"text": "To find the expected payoff in a game of chance, list all possible outcomes, determine the value and probability of each, and calculate the weighted average. This tells you the average result you can expect if you play the game many times.",
"audio": "To find the expected payoff in a game of chance, list all possible outcomes, determine the value and probability of each, and calculate the weighted average. This tells you the average result you can expect if you play the game many times."
},
"scenes": [
{
"text": "What is expected payoff? It\u2019s the average amount you can expect to win or lose per game if you played it many times. It\u2019s a weighted average of all outcomes, where each outcome is weighted by its probability.",
"latex": "E = P_1 \\times V_1 + P_2 \\times V_2 + \\ldots + P_n \\times V_n",
"pop_animation_prompt": "Create pop animations for the mathematical expression. First, pop the elements \"P_1\", \"P_2\", and \"P_n\" simultaneously when the word \"probability\" is mentioned in the transcript. Ensure these elements are highlighted together to emphasize their role in the expression."
},
{
"text": "How do you calculate it? You need two things for each outcome, the value of that outcome, like how much you win or lose, and the probability of that outcome. Then you use the formula, E, equals the sum from i queals 1, to n of P, i times V, i. The total of all probabilities must be 1.",
"latex": "V= \\text{Value} \\quad P = \\text{Probability} \\\\ E = \\text{sum}_{i=1}^{n} P_i V_i \\quad \\text{where } \\text{sum} \\ P_i = 1",
"pop_animation_prompt": "Create pop animations for the following elements as they are referenced in the transcript: First, pop \"E\" and \"=\" together. Next, pop the \"sum\" expression, followed by \"i\", \"equals\", \"1\", and \"n\" in sequence. Then, pop \"P\" and \"V\" with their subscripts \"i\" simultaneously. Finally, pop the \"probabilities\" expression and the number \"1\" together. Ensure each pop aligns with its mention in the transcript for clarity and emphasis."
},
{
"text": "For a simple example, you toss a fair coin. If it's heads, you win $10. If it's tails, you win nothing. Both outcomes are equally likely. The expected payoff is, 0.5 times 10, plus 0.5 times 0, which equals 5. So, on average, you win $5 per toss if you play many times.",
"latex": "E = (0.5 \\times 10) + (0.5 \\times 0) = 5",
"pop_animation_prompt": "Create pop animations for the mathematical expression. First, pop the entire expression when \"expected\" is mentioned. Then, sequentially pop \"0.5\" , \"10\" , \"0.5\" , and \"0\" as they are described. Finally, pop the result \"5\" when \"$5\" is mentioned."
},
{
"text": "For another example, you spin a wheel with three equal sections, Red, Blue, and Green. It costs $2 to play. If you land on Red, you win $5, so the net gain is $3. On Blue, you win $1, so you lose $1 overall. On Green, you win nothing and lose the full $2. Each outcome has a probability of one-third.",
"latex": "V_{Red}=3, \\ V_{Blue}=-1, \\ V_{Green}=-2 \\\\ P=\\frac{1}{3} \\text{ for all}",
"pop_animation_prompt": "Create pop animations for the following elements as they are referenced in the manuscript: First, pop \"V_{Red}\" and \"V_{Blue}\" and \"V_{Green}\" together. Then, pop \"V_{Red}\" again, followed by \"3\". Next, pop \"V_{Blue}\" and then \"-1\". After that, pop \"V_{Green}\" and then \"-2\". Finally, pop the fraction \"\\frac{1}{3}\". Ensure each element pops at the time it is mentioned in the transcript."
},
{
"text": "Now, to calculate the spinner\u2019s expected payoff, you find that E, equals one-third times 3, plus one-third times minus 1, plus one-third times minus 2. That equals 1 minus one-third minus two-thirds, which is 0. An expected payoff of zero means the game is fair, you don\u2019t win or lose money on average.",
"latex": "E = \\frac{1}{3} \\times 3 + \\frac{1}{3} \\times (-1) + \\frac{1}{3} \\times (-2) = 1 - \\frac{1}{3} - \\frac{2}{3} = 0",
"pop_animation_prompt": "Create pop animations for the following math elements as they are referenced: \"E\" and \"=\" together, then \"\\(\\frac{1}{3}\\) and \"\u00d7\" and \"3\" together, followed by \"+\" and \"\\(\\frac{1}{3}\\) and \"\u00d7\" and \"-1\" together, then \"+\" and \"\\(\\frac{1}{3}\\) and \"\u00d7\" and \"-2\" together. Next, \"=\" and \"1\" together, followed by \"-\" and \"\\(\\frac{1}{3}\\) and \"-\" and \"\\(\\frac{2}{3}\\) together, and finally \"=\" and \"0\" together. Ensure each group pops at the time it is mentioned in the transcript."
},
{
"text": "How do you interpret the expected payoff? If E is greater than zero, the game is favorable to the player. If E is less than zero, the game is unfavorable, you expect to lose money on average. If E, equals zero, the game is fair. And remember, this is the long-run average, not what happens in a single play.",
"latex": "E > 0: \\text{Favorable} \\quad | \\quad E < 0: \\text{Unfavorable} \\quad | \\quad E = 0: \\text{Fair}",
"pop_animation_prompt": "Create pop animations for the following elements as they are referenced in the transcript: First, pop \"E\" (S6.e), \">\" (S6.operator), and \"0\" (S6.0) together, followed by \"Favorable\" (S6.favorable). Next, pop \"E\" (S6.e.4), \"<\" (S6.operator.5), and \"0\" (S6.0.3) together, followed by \"Unfavorable\" (S6.unfavorable). Finally, pop \"E\" (S6.e.6), \"=\" (S6.operator.8), and \"0\" (S6.0.4) together, followed by \"Fair\" (S6.fair)."
}
],
"outro": {
"text": "To find the expected payoff in a game of chance, list all possible outcomes, assign a value and probability to each, and calculate the weighted average. This gives you the average result you can expect if you play the game many times.",
"audio": "To find the expected payoff in a game of chance, list all possible outcomes, assign a value and probability to each, and calculate the weighted average. This gives you the average result you can expect if you play the game many times."
}
}
}
en_12_prob_exp_payoff_game_of_chance.json
Displaying en_12_prob_exp_payoff_game_of_chance.json.
