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Chapter 4: Problem 11

Is it Bernoulli? Determine if each trial can be considered an independent Bernoulli trial for the following situations. (a) Cards dealt in a hand of poker. (b) Outcome of each roll of a die.

Short Answer

Expert verified
(a) No, not Bernoulli; (b) Yes, Bernoulli.

Step by step solution

01

Understanding Bernoulli Trials

A Bernoulli trial is an experiment with exactly two possible outcomes, typically termed 'success' and 'failure'. Each trial is independent, meaning the outcome of one does not affect the other.
02

Analyzing Situation (a): Cards dealt in a hand of poker

In a hand of poker, a standard deck of cards is used. As cards are dealt, the probability of drawing specific cards changes because the deck size decreases. Thus, the trials are not independent鈥攊f you know a card has been dealt, this affects the probabilities for subsequent cards. Therefore, this situation is not a Bernoulli trial.
03

Analyzing Situation (b): Outcome of each roll of a die

When rolling a die, each roll is independent because the result of one roll does not affect the outcome of any other roll. Each has the same chance (1/6 for a six-sided die) of landing on each face, which remains constant regardless of previous rolls. Therefore, this situation is a Bernoulli trial.

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Key Concepts

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

Independent Trials
In probability theory, independent trials are fundamental to understanding random experiments like Bernoulli trials. An experiment is deemed independent when the outcome of one trial does not influence or alter the outcome of another. Independence is crucial since it assures that the probability of outcomes remains consistent across trials.
  • In the context of dice rolling, each roll is independent. It does not matter what results from a previous roll; the next roll is completely unaffected.
  • In contrast, a hand of cards dealt from a single deck is not independent, since each card removed changes the possible outcomes for the next draw.
Understanding whether a trial is independent helps determine if it fits the Bernoulli trial criteria, which demand independent events for valid analysis.
Probability
Probability measures how likely an event is to occur. It is a number between 0 and 1, where 0 indicates impossibility and 1 certainty. In Bernoulli trials, the probability helps define the chance of 'success' and 'failure'.
  • For a fair six-sided die, the probability of rolling any specific number (like a 3) is \( \frac{1}{6} \), due to the equal likelihood of each face landing up.
  • In contrast, when dealing cards from a deck, probabilities change with each draw, since the number of remaining cards decreases, affecting the likelihood of drawing a specific card on subsequent deals.
Understanding probability within this framework is vital for correctly analyzing any situation as possible Bernoulli experiments.
Success and Failure Outcomes
Every Bernoulli trial is designed around two possible outcomes, commonly labeled as 'success' and 'failure'. This binary classification helps facilitate analysis and calculations in probability.
  • In dice rolling, defining 'success' could be rolling a number greater than 4, while 'failure' could be rolling a number less than or equal to 4.
  • In poker, 'success' might mean drawing a face card, while 'failure' may mean drawing any other card.
Clearly outlining what constitutes 'success' and 'failure' is essential to applying Bernoulli principles accurately in real-world scenarios.
Poker Hand Analysis
Poker hand analysis can be quite complex, especially since cards are not drawn independently in a standard game. As such, poker does not fulfill the criteria for a Bernoulli trial because of the dependency between draws. In poker:
  • As cards are dealt, the deck's composition changes, influencing the probabilities of drawing certain cards next.
  • This situation means each draw affects subsequent draws' odds, breaking the independence condition required for Bernoulli trials.
Therefore, while poker offers many interesting probability problems, it does not fit the Bernoulli model.
Die Roll Probability
Rolling a die is often used as a classic example of independent and random probability experiments fitting the Bernoulli trial model.When you roll a die:
  • Each roll is an independent event; previous results do not influence future outcomes.
  • Every face has an equal chance (if the die is fair), meaning the probability stays constant across trials 鈥� \( \frac{1}{6} \) chance per face.
This characteristic makes dice rolling a perfect example of Bernoulli trials, showcasing both independence in action and the consistent application of probability.

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Most popular questions from this chapter

Serving in volleyball. A not-so-skilled volleyball player has a \(15 \%\) chance of making the serve, which involves hitting the ball so it passes over the net on a trajectory such that it will land in the opposing team's court. Suppose that her serves are independent of each other. (a) What is the probability that on the \(10^{\text {th }}\) try she will make her \(3^{r d}\) successful serve? (b) Suppose she has made two successful serves in nine attempts. What is the probability that her \(10^{t h}\) serve will be successful? (c) Even though parts (a) and (b) discuss the same scenario, the probabilities you calculated should be different. Can you explain the reason for this discrepancy?

Triathlon times, Part I. In triathlons, it is common for racers to be placed into age and gender groups. Friends Leo and Mary both completed the Hermosa Beach Triathlon, where Leo competed in the Men, Ages 30 - 34 group while Mary competed in the Women, Ages 25 - 29 group. Leo completed the race in 1:22:28 (4948 seconds), while Mary completed the race in 1:31:53 (5513 seconds). Obviously Leo finished faster, but they are curious about how they did within their respective groups. Can you help them? Here is some information on the performance of their groups: \- The finishing times of the Men, Ages 30 - 34 group has a mean of 4313 seconds with a standard deviation of 583 seconds. \- The finishing times of the Women, Ages \(25-29\) group has a mean of 5261 seconds with a standard deviation of 807 seconds. \- The distributions of finishing times for both groups are approximately Normal. Remember: a better performance corresponds to a faster finish. (a) Write down the short-hand for these two normal distributions. (b) What are the Z-scores for Leo's and Mary's finishing times? What do these Z-scores tell you? (c) Did Leo or Mary rank better in their respective groups? Explain your reasoning. (d) What percent of the triathletes did Leo finish faster than in his group? (e) What percent of the triathletes did Mary finish faster than in her group? (f) If the distributions of finishing times are not nearly normal, would your answers to parts (b) - (e) change? Explain your reasoning.

Underage drinking, Part II. We learned in Exercise 4.17 that about \(70 \%\) of \(18-20\) year olds consumed alcoholic beverages in any given year. We now consider a random sample of fifty \(18-20\) year olds. (a) How many people would you expect to have consumed alcoholic beverages? And with what standard deviation? (b) Would you be surprised if there were 45 or more people who have consumed alcoholic beverages? (c) What is the probability that 45 or more people in this sample have consumed alcoholic beverages? How does this probability relate to your answer to part (b)?

Customers at a coffee shop. A coffee shop serves an average of 75 customers per hour during the morning rush. (a) Which distribution have we studied that is most appropriate for calculating the probability of a given number of customers arriving within one hour during this time of day? (b) What are the mean and the standard deviation of the number of customers this coffee shop serves in one hour during this time of day? (c) Would it be considered unusually low if only 60 customers showed up to this coffee shop in one hour during this time of day? (d) Calculate the probability that this coffee shop serves 70 customers in one hour during this time of day.

Area under the curve, Part II. What percent of a standard normal distribution \(N(\mu=0, \sigma=1)\) is found in each region? Be sure to draw a graph. (a) \(Z>-1.13\) (b) \(Z<0.18\) (c) \(Z>8\) (d) \(|Z|<0.5\)
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