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Consider the following games with two dice. a. A gambler is going to roll a die four times. If he rolls at least one 6, you must pay him \(\$ 5 .\) If he fails to roll a 6 in four tries, he will pay you \(\$ 5\). Find the probability that you must pay the gambler. Assume that there is no cheating. b. The same gambler offers to let you roll a pair of dice 24 times. If you roll at least one double 6 , he will pay you \(\$ 10\). If you fail to roll a double 6 in 24 tries, you will pay him \(\$ 10\). The gambler says that you have a better chance of winning because your probability of success on each of the 24 rolls is \(1 / 36\) and you have 24 chances. Thus, he says, your probability of winning \(\$ 10\) is \(24(1 / 36)=2 / 3\). Do you agree with this analysis? If so, indicate why. If not, point out the fallacy in his argument, and then find the correct probability that you will win.

Step by step solution

01

Part A: Calculate the Probability of Rolling At Least One Six in Four Tries.

The most straightforward way of figuring out the probability that the dice shows at least one 6 in four throws is to determine the probability that no 6 appears in four throws and subtract this probability from 1. Each throw is independent, and the chance of not getting a 6 is \(5 / 6\). Therefore, the possibility of not getting a 6 in four tries is \((5 / 6) ^ 4\). Hence, the probability of getting at least one 6 is \(1 - (5 / 6) ^ 4\).

02

Part B: Evaluate the Gambler's Probability Calculation for Rolling Double Sixes 24 Times.

To start with, we should point out the mistake in the gambler’s argument. The dice rolls are independent events; thus, the sum of each individual event's probabilities doesn't equal the total probability. Indeed, probability multiplication is only valid for independent events, which is not the case here.

03

Part B: Find the Correct Probability of Rolling Double Sixes 24 Times.

Similar to part A, it's much easier to calculate the probability of the opposite event of not rolling double sixes within 24 tries. The probability of not getting a double six in one roll is \(35 / 36\). Consequently, the probability of not rolling double sixes 24 times is \((35 / 36) ^ {24}\). Hence, the probability of getting at least one double six during 24 rolls is \(1 - (35 / 36) ^ {24}\).

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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, independent events are events that do not influence each other. This means the occurrence of one event does not affect the probability of the occurrence of another. For instance, if you're rolling a die, each roll is an independent event. The result of one roll does not change the outcomes of subsequent rolls.
This independence is crucial when calculating the probability of multiple events happening in sequence. You multiply the probabilities of each individual event.
For example, the chance of NOT rolling a 6 on a single die is \( \frac{5}{6} \). Since each roll doesn't affect the others, for four rolls, the probability of never rolling a 6 is \( \left( \frac{5}{6} \right)^4 \). Therefore, independent events especially impact how we calculate combined probabilities.

Probability Calculations

Probability calculations help us understand the likelihood of different outcomes. In probability, we often use the formula for complementary events to simplify our calculations.
The probability of not rolling a 6 in one throw of a die is \( \frac{5}{6} \). To find the probability of at least one 6 in four rolls, we first find the probability of the complementary event, which is not rolling any 6's. Then, we subtract this from 1.
This gives us \( 1 - \left( \frac{5}{6} \right)^4 \) for rolling at least one 6. This approach is essentially the same when dealing with complex calculations, such as determining the chance of rolling at least one double 6 in 24 dice rolls.

Rolling Dice

Rolling dice is a common example used in probability theory because each die has an equal chance of landing on any one of its six sides. This uniform probability distribution makes calculations straightforward.
When rolling a pair of dice, the probability of rolling a double 6 (getting a 6 on both dice) is \( \frac{1}{36} \).
However, calculating probabilities for multiple attempts, such as rolling the pair 24 times, it's easier to consider the complementary event— not rolling a double 6. This probability is \( \frac{35}{36} \) for each roll. To find the probability of rolling at least one double 6 in 24 attempts, we calculate \[ 1 - \left( \frac{35}{36} \right)^{24} \].
Understanding these basics of dice and probability allows us to tackle a wide variety of problems in probability theory.