91Ó°ÊÓ

The geometric distributions You are tossing a pair of fair, six-sided dice in a board game. Tosses are independent. You land in a danger zone that requires you to roll doubles (both faces showing the same number of spots) before you are allowed to play again. How long will you wait to play again? (a) What is the probability of rolling doubles on a single toss of the dice? (If you need review, the possible outcomes appear in Figure 5.2 (page 300). All 36 outcomes are equally likely.) (b) What is the probability that you do not roll doubles on the first toss, but you do on the second toss? (c) What is the probability that the first two tosses are not doubles and the third toss is doubles? This is the probability that the first doubles occurs on the third toss. (d) Now you see the pattern. What is the probability that the first doubles occurs on the fourth toss? On the fifth toss? Give the general result: what is the probability that the first doubles occurs on the kth toss?

Step by step solution

01

Calculate Probability of Rolling Doubles

Each die has 6 faces, so there are \( 6 \times 6 = 36 \) total possible outcomes when rolling two dice. The doubles occur when both dice show the same number, which can be 1-1, 2-2, ..., 6-6. There are 6 possible double outcomes. Since each outcome is equally likely, the probability of rolling doubles on a single toss is \( \frac{6}{36} = \frac{1}{6} \).

02

Calculate Probability for Doubles on Second Toss

To roll doubles on the second toss, the first toss must result in non-doubles. The probability of non-doubles on the first toss is \( 1 - \frac{1}{6} = \frac{5}{6} \). The probability of rolling doubles on the second toss is \( \frac{1}{6} \). Therefore, the probability of not rolling double on the first toss but rolling it on the second toss is \( \frac{5}{6} \times \frac{1}{6} = \frac{5}{36} \).

03

Calculate Probability for Doubles on Third Toss

For the first doubles to occur on the third toss, the first and second tosses must be non-doubles, and the third must be doubles. The probability is: \(\left( \frac{5}{6} \right)^2 \times \frac{1}{6} = \frac{25}{216} \).

04

Calculate General Probability for Doubles on k-th Toss

Similarly, if the first doubles occurs on the \( k \)-th toss, the first \( (k-1) \) tosses must be non-doubles, and the \( k \)-th toss must be doubles. The probability is given by the formula: \[P(\text{first doubles on } k\text{-th toss}) = \left( \frac{5}{6} \right)^{k-1} \times \frac{1}{6}\]

05

Apply Formula for Fourth and Fifth Tosses

Using the general formula from Step 4:- For the fourth toss: \( \left( \frac{5}{6} \right)^3 \times \frac{1}{6} = \frac{125}{1296} \).- For the fifth toss: \( \left( \frac{5}{6} \right)^4 \times \frac{1}{6} = \frac{625}{7776} \).

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Key Concepts

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

Probability

Probability is a way to measure how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 means the event will not happen and 1 means the event is certain to occur. In the context of rolling dice, each possible outcome of rolling two dice is equally likely. This simple scenario helps us calculate the probability of rolling doubles, i.e., when both dice show the same number.

• We have 36 possible outcomes when rolling two six-sided dice (since each die has 6 faces: 6 x 6 = 36).
• Doubles can occur in 6 ways: 1-1, 2-2, 3-3, 4-4, 5-5, and 6-6.

Thus, the probability of rolling doubles in one toss is the number of favorable outcomes (doubles) divided by the total number of possible outcomes. That means for doubles, the probability is \( \frac{1}{6} \). Understanding this probability helps in calculating the chances of doubles occurring on subsequent rolls.

Independent Events

In probability theory, events are said to be independent if the occurrence of one does not affect the occurrence of another. With rolling dice, each roll is independent, meaning that the outcome of one toss doesn’t affect the outcome of the next toss.

• This means, for example, getting a double on the first toss does not influence getting a double on the second toss.
• Because each roll is independent, the probability of any outcome on a dice roll is always the same, irrespective of previous outcomes.

For example, even if you do not roll doubles on the first two tosses, the probability of getting doubles on the third toss still remains \( \frac{1}{6} \). This characteristic is crucial in determining the probability of rolling doubles for the first time on any given roll.

Rolling Dice

Rolling dice is one of the simplest random events used to explain probability concepts. When you roll two dice, each die faces an independent roll wherein each face from 1 to 6 has an equal chance of appearing.

• Each die in a two-dice roll has 6 possible outcomes, creating a total of 36 combined outcomes.
• The act of rolling these two dice together helps in studying probabilities like rolling specific combinations, such as doubles.

This setup is perfect for learning about independent events, probability distribution, and calculating the likelihood of various events. Understanding how dice rolls work, including how likely it is to roll doubles, sets the foundation for more complex probability problems.

Probability Distribution

A probability distribution informs us about all possible outcomes of an event and their corresponding probabilities. In rolling dice, especially when looking out for doubles, we create a probability distribution by focusing on when the first successful double appears after the nth roll.

• We start by calculating the probability of not rolling doubles (\( \frac{5}{6} \)).
• Then, we calculate the probability of rolling a double on specific rolls, like the 2nd roll or 3rd roll, using the geometric distribution.

For example, to find the probability of rolling doubles for the first time on the kth roll, the formula is:\[ P(\text{first doubles on } k\text{-th toss}) = \left( \frac{5}{6} \right)^{k-1} \times \frac{1}{6} \]This distribution helps understand the probability of occurrences and is vital in many statistical calculations and real-world applications.