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Chapter 1: Problem 12

You offer 3: 1 odds that your friend Smith will be elected mayor of your city. What probability are you assigning to the event that Smith wins?

Short Answer

Expert verified
The probability that Smith will win is \( \frac{3}{4} \) or 75%.

Step by step solution

01

Understand Odds

Odds represent the ratio of the probability that an event will occur to the probability that it will not occur. Here, the odds 3:1 mean there are 3 chances that Smith will win compared to 1 chance Smith will not win.
02

Set Up the Probability Equation

If the odds of winning are 3:1, this means that out of 4 total outcomes (3 + 1), Smith is expected to win 3 of them. The formula for probability is the number of favorable outcomes divided by the total number of outcomes.
03

Calculate the Probability

The probability can be calculated using the formula: \( P( ext{Smith wins}) = \frac{ ext{Number of favorable outcomes}}{ ext{Total number of outcomes}} = \frac{3}{3+1} = \frac{3}{4} \).
04

Convert to Percentage (Optional)

To express the probability as a percentage, multiply the probability by 100. Thus, \( \frac{3}{4} = 0.75 \), which translates to a 75% chance that Smith will win.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Odds
Odds are a way of expressing the likelihood of an event happening compared to it not happening. They provide a comparative measure of two scenarios, often written in a ratio format. For example, the odds 3:1 indicate there are 3 chances of the event happening for every 1 chance of it not happening.
To dissect this concept:
  • A ratio of 3:1 means for every 4 outcomes, 3 are favorable and 1 is not.
  • Odds highlight the contrast between success and failure of an event.
This comparative nature helps in quickly understanding the situation without diving deep into specific probabilities.
Deciphering Favorable Outcomes
A favorable outcome refers to a result that aligns with the prediction or hope regarding an event. When we discuss the odds of 3:1, it implies:
  • There are 3 favorable outcomes.
  • These are compared against just 1 unfavorable outcome (not winning in this case).
The total number of outcomes is the sum of favorable and unfavorable outcomes, which is crucial for probability calculation. Here, with a total of 4 outcomes (3 favorable + 1 unfavorable), we bridge these odds into a meaningful probability.
Steps of Probability Calculation
Probability provides a numerical measure of the chance that an event will occur. Calculating probability involves determining the proportion of favorable outcomes to the total possible outcomes.
The probability formula is as follows:
  • Given odds of 3:1,
  • The probability (P) that Smith wins equals:
  • \[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{3}{4} \]
A 3 out of 4 probability indicates a high likelihood of Smith winning, showcasing how odds translate into a tangible probability.
Converting Probability to Percentage
Converting a probability into a percentage makes the data more comprehensible and relatable.
To convert the probability fraction to a percentage:
  • Multiply the fraction by 100.
  • For example, \[ \frac{3}{4} \times 100 = 75 \].
This calculation results in a 75% probability that Smith will win. Expressing probability in percentage form is beneficial for easier interpretation, especially when communicating findings to those more familiar with percentage-based chances.

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Most popular questions from this chapter

Modify the program CoinTosses so that it tosses a coin \(n\) times and records whether or not the proportion of heads is within .1 of .5 (i.e., between .4 and .6 ). Have your program repeat this experiment 100 times. About how large must \(n\) be so that approximately 95 out of 100 times the proportion of heads is between .4 and \(.6 ?\)

A student must choose one of the subjects, art, geology, or psychology, as an elective. She is equally likely to choose art or psychology and twice as likely to choose geology. What are the respective probabilities that she chooses art, geology, and psychology?

Let \(A\) and \(B\) be events such that \(P(A \cap B)=1 / 4, P(\tilde{A})=1 / 3,\) and \(P(B)=\) \(1 / 2 .\) What is \(P(A \cup B) ?\)

A die is loaded in such a way that the probability of each face turning up is proportional to the number of dots on that face. (For example, a six is three times as probable as a two.) What is the probability of getting an even number in one throw?

You are offered the following game. A fair coin will be tossed until the first time it comes up heads. If this occurs on the \(j\) th toss you are paid \(2^{j}\) dollars. You are sure to win at least 2 dollars so you should be willing to pay to play this game - but how much? Few people would pay as much as 10 dollars to play this game. See if you can decide, by simulation, a reasonable amount that you would be willing to pay, per game, if you will be allowed to make a large number of plays of the game. Does the amount that you would be willing to pay per game depend upon the number of plays that you will be allowed?
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