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Problem 1 Consider the random trial of rolling a fair die until a 6 is rolled, ...

Sep 10, 2024

Problem 1 Consider the random trial of rolling a fair die until a 6 is rolled, and counting how many tries this takes. a) What is the sample space? b) Given a positive integer n, what is the probability it take n tries to roll and n? c) Define the event A = “it takes an odd number of tries”. Find P(A).

Solution by Steps

step 1

The sample space for rolling a fair die consists of all possible outcomes when rolling the die. The possible outcomes are:

\{1, 2, 3, 4, 5, 6\}

. Thus, the sample space is

S = \{1, 2, 3, 4, 5, 6\}

with a total of 6 outcomes

step 2

The probability of rolling a 6 on the nth try can be modeled as a geometric distribution. The probability of success (rolling a 6) is

p = \frac{1}{6}

and the probability of failure (not rolling a 6) is

q = \frac{5}{6}

. The probability that it takes exactly

n

tries to roll a 6 is given by:

P(X = n) = q^{n-1} \cdot p = \left(\frac{5}{6}\right)^{n-1} \cdot \frac{1}{6}

step 3

To find the probability

P(A)

that it takes an odd number of tries to roll a 6, we sum the probabilities for all odd

n

P(A) = P(X = 1) + P(X = 3) + P(X = 5) + \ldots = \sum_{k=0}^{\infty} P(X = 2k+1) = \sum_{k=0}^{\infty} \left(\frac{5}{6}\right)^{2k} \cdot \frac{1}{6}

This is a geometric series with first term

\frac{1}{6}

and common ratio

\left(\frac{5}{6}\right)^2

step 4

The sum of the infinite geometric series is given by:

S = \frac{a}{1 - r} = \frac{\frac{1}{6}}{1 - \left(\frac{5}{6}\right)^2} = \frac{\frac{1}{6}}{1 - \frac{25}{36}} = \frac{\frac{1}{6}}{\frac{11}{36}} = \frac{6}{11}

Thus,

P(A) = \frac{6}{11}

Answer

The sample space is

S = \{1, 2, 3, 4, 5, 6\}

, the probability of rolling a 6 on the nth try is

P(X = n) = \left(\frac{5}{6}\right)^{n-1} \cdot \frac{1}{6}

, and

P(A) = \frac{6}{11}

Key Concept

The concept of probability in a geometric distribution, particularly in the context of rolling a die until a specific outcome occurs.

Explanation

The solution involves understanding the sample space, calculating the probability of a specific outcome on a given trial, and summing probabilities for an infinite series to find the probability of an event occurring in an odd number of tries.