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Chapter 3: Problem 4

Backgammon is a board game for two players in which the playing pieces are moved according to the roll of two dice. Players win by removing all of their pieces from the board, so it is usually good to roll high numbers. You are playing backgammon with a friend and you roll two 6 s in your first roll and two 6 s in your second roll. Your friend rolls two \(3 \mathrm{~s}\) in his first roll and again in his second row. Your friend claims that you are cheating, because rolling double 6 s twice in a row is very unlikely. Using probability, show that your rolls were just as likely as his.

Short Answer

Expert verified
Your rolls were as likely as your friend's, both having a probability of 1/1296.

Step by step solution

01

Calculate the Probability of Rolling Double Sixes

To find the probability of rolling two 6s with two dice, observe that the probability of one die showing a 6 is \(\frac{1}{6}\). To have both dice show 6, multiply the probabilities: \(\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}\). Therefore, the probability for rolling double 6s is \(\frac{1}{36}\).
02

Calculate the Probability of Rolling Double Sixes Twice

Since each roll of two dice is independent, to roll double 6s twice in a row, multiply the probabilities of each occurrence. This is \(\frac{1}{36} \times \frac{1}{36} = \frac{1}{1296}\).
03

Calculate the Probability of Rolling Double Threes

Similar to double 6s, the probability of rolling two 3s is also \(\frac{1}{36}\) because each die needs to show a 3: \(\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}\).
04

Calculate the Probability of Rolling Double Threes Twice

As with the double 6s calculation, the probability of rolling two 3s twice consecutively is \(\frac{1}{36} \times \frac{1}{36} = \frac{1}{1296}\).
05

Compare the Probabilities

Both rolling double 6s twice and rolling double 3s twice have the same probability, \(\frac{1}{1296}\). Therefore, both events are equally likely.

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Key Concepts

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

Independent Events
In probability theory, an independent event is one whose outcome does not influence the outcome of another event. This means that the occurrence of one event does not affect the likelihood of another event occurring. Consider rolling a pair of dice: the result of the first roll does not influence the second roll. Each roll is its separate event.
  • For independent events, the probability of both events occurring is calculated by multiplying their probabilities.
  • For example, rolling a double six in the first roll is independent of rolling double six again; thus, you can multiply their probabilities.
This independence principle is foundational in understanding how probabilities accumulate in a sequence of events.
Dice Probability
Dice probability deals with the chances of achieving a specific outcome when rolling dice. Each die has six faces numbered from 1 to 6. For a single die, the probability of landing on any one particular number is 1 out of 6, or \( \frac{1}{6} \).
  • When rolling two dice, the objective often involves determining the joint outcome, such as obtaining a particular sum or double numbers.
  • Calculating the probability of getting double sixes involves one die showing a six, followed by the second die also showing a six.
This results in a probability calculation of \( \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \), showing how rarity increases with specific criteria.
Probability Calculation
Probability calculations are fundamental to predicting the likelihood of events occurring, especially in games of chance. To calculate probabilities effectively:
  • Identify the event's desired outcome and all possible outcomes.
  • Divide the number of desired outcomes by the total possible outcomes. This fraction represents the probability.
For example, calculating the chance of rolling double sixes twice requires determining the initial probability of rolling two sixes (\( \frac{1}{36} \) per roll). Since these are independent events (as detailed before), multiply the probability of the first occurrence by the probability of the second occurrence: \( \frac{1}{36} \times \frac{1}{36} = \frac{1}{1296} \). This helps understand not just singular possibilities but their sequences.
Probability Comparison
Probability comparison involves evaluating the likelihood of different events happening to see which is more probable. In many cases, such as games involving dice, it's insightful to compare these probabilities to make informed decisions.
  • For instance, in backgammon, both rolling double sixes twice and rolling double threes twice each have a probability of \( \frac{1}{1296} \).
  • By comparing these equal probabilities, we can conclude that both outcomes are equally plausible in terms of chance.
This type of analysis helps quickly dispel myths or assumptions about one series of events being more likely than another, reinforcing a logical and mathematical understanding.

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Most popular questions from this chapter

Assortative mating is a nonrandom mating pattern where individuals with similar genotypes and/or phenotypes mate with one another more frequently than what would be expected under a random mating pattern. Researchers studying this topic collected data on eye colors of 204 Scandinavian men and their female partners. The table below summarizes the results. For simplicity, we only include heterosexual relationships in this exercise. \(^{42}\) $$\begin{array}{lcccc} & {\text { Partner (female) }} & \\ & \text { Blue } & \text { Brown } & \text { Green } & \text { Total } \\ \hline \text { Blue } & 78 & 23 & 13 & 114 \\ \text { Brown } & 19 & 23 & 12 & 54 \\ \text { Green } & 11 & 9 & 16 & 36 \\ \hline \text { Total } & 108 & 55 & 41 & 204\end{array}$$ (a) What is the probability that a randomly chosen male respondent or his partner has blue eyes? (b) What is the probability that a randomly chosen male respondent with blue eyes has a partner with blue eyes? (c) What is the probability that a randomly chosen male respondent with brown eyes has a partner with blue eyes? What about the probability of a randomly chosen male respondent with green eyes having a partner with blue eyes? (d) Does it appear that the eye colors of male respondents and their partners are independent? Explain your reasoning.

The game of roulette involves spinning a wheel with 38 slots: 18 red, 18 black, and 2 green. A ball is spun onto the wheel and will eventually land in a slot, where each slot has an equal chance of capturing the ball. (a) You watch a roulette wheel spin 3 consecutive times and the ball lands on a red slot each time. What is the probability that the ball will land on a red slot on the next spin? (b) You watch a roulette wheel spin 300 consecutive times and the ball lands on a red slot each time. What is the probability that the ball will land on a red slot on the next spin? (c) Are you equally confident of your answers to parts (a) and (b)? Why or why not?

A Pew Research survey asked 2,373 randomly sampled registered voters their political affiliation (Republican, Democrat, or Independent) and whether or not they identify as swing voters. \(35 \%\) of respondents identified as Independent, \(23 \%\) identified as swing voters, and \(11 \%\) identified as both. \(^{21}\) (a) Are being Independent and being a swing voter disjoint, i.e. mutually exclusive? (b) Draw a Venn diagram summarizing the variables and their associated probabilities. (c) What percent of voters are Independent but not swing voters? (d) What percent of voters are Independent or swing voters? (e) What percent of voters are neither Independent nor swing voters? (f) Is the event that someone is a swing voter independent of the event that someone is a political Independent?

A Pew Research poll asked 1,306 Americans "From what you've read and heard, is there solid evidence that the average temperature on earth has been getting warmer over the past few decades, or not?". The table below shows the distribution of responses by party and ideology, where the counts have been replaced with relative frequencies. $$\begin{array}{lcccc} &{\text { Response }} & \\ & \begin{array}{c}\text { Earth is } \\ \text { warming }\end{array} & \begin{array}{c}\text { Not } \\\\\text { warming } \end{array} & \begin{array}{c}\text { Don't Know } \\\\\text { Refuse }\end{array} & \text { Total } \\ \hline \text { Conservative Republican } & 0.11 & 0.20 & 0.02 & 0.33 \\ \text { Mod/Lib Republican } & 0.06 & 0.06 & 0.01 & 0.13 \\ \text { Mod/Cons Democrat } & 0.25 & 0.07 & 0.02 & 0.34 \\ \text { Liberal Democrat } & 0.18 & 0.01 & 0.01 & 0.20 \\ \hline \text { Total } & 0.60 & 0.34 & 0.06 & 1.00 \end{array}$$ (a) Are believing that the earth is warming and being a liberal Democrat mutually exclusive? (b) What is the probability that a randomly chosen respondent believes the earth is warming or is a liberal Democrat? (c) What is the probability that a randomly chosen respondent believes the earth is warming given that he is a liberal Democrat? (d) What is the probability that a randomly chosen respondent believes the earth is warming given that he is a conservative Republican? (e) Does it appear that whether or not a respondent believes the earth is warming is independent of their party and ideology? Explain your reasoning. (f) What is the probability that a randomly chosen respondent is a moderate/liberal Republican given that he does not believe that the earth is warming?
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