Problem 11
Question
You decide to play monthly in two different lotteries, and you stop playing as soon as you win a prize in one (or both) lotteries of at least one million euros. Suppose that every time you participate in these lotteries, the probability to win one million (or more) euros is \(p_{1}\) for one of the lotteries and \(p_{2}\) for the other. Let \(M\) be the number of times you participate in these lotteries until winning at least one prize. What kind of distribution does \(M\) have, and what is its parameter?
Step-by-Step Solution
Verified Answer
\(M\) follows a geometric distribution with parameter \( p = 1 - (1 - p_1)(1 - p_2) \).
1Step 1: Understand the Problem Statement
You are participating in two lotteries each month, and you stop playing as soon as you win a prize of at least one million euros in either lottery. Each lottery has its own probability of winning, denoted as \( p_1 \) and \( p_2 \). We need to find the distribution and parameter for \( M \), which is the number of times you participate until winning.
2Step 2: Recognize the Event Type
Since you continue to participate until a success occurs (winning at least one of the lotteries), this scenario describes a series of independent Bernoulli trials, which typically follows a geometric distribution when considering the trial at which the first success occurs.
3Step 3: Determine the Success Probability for Each Attempt
The probability of success in any given month is the probability of winning at least one of the lotteries. The probability of not winning in a given lottery is \( 1 - p_1 \) for the first and \( 1 - p_2 \) for the second. Thus, the probability of not winning in both lotteries is \((1 - p_1) \times (1 - p_2)\). The probability of winning at least one is thus the complement: \[ p = 1 - (1 - p_1)(1 - p_2) \]
4Step 4: Identify the Distribution of M
Since \( M \) is the number of participations until the first winning event occurs, and each participation is an independent trial, \( M \) follows a geometric distribution. The geometric distribution models the number of trials up to and including the first success, based on the constant success probability \( p \).
5Step 5: State the Parameter of the Distribution
The parameter of the geometric distribution is the probability of success per trial, which we have already identified as \( p = 1 - (1 - p_1)(1 - p_2) \). Hence, the random variable \( M \) follows a geometric distribution with parameter \( p \).
Key Concepts
Bernoulli TrialsProbability of WinningSuccess ProbabilityParameter Calculation
Bernoulli Trials
To understand the exercise, it's important to grasp what Bernoulli trials are. A Bernoulli trial is a random experiment where there are exactly two possible outcomes: success or failure. These trials are named after the mathematician Jakob Bernoulli. In the context of the exercise, each time you play both lotteries is considered one trial.
For each trial, you check if the outcome is a success (winning at least one of the lotteries) or a failure (winning none). Because each trial is conducted independently, the outcome of one trial does not affect the others. This means that if you win this month, the chances of winning next month remain the same. This independence is key for the trials to fall under the Bernoulli trial category.
Bernoulli trials form the foundational concept that leads into the geometric distribution, which tells us the probability of achieving a success within a number of trials. This concept is crucial for solving the problem and understanding the role of lottery participation as a sequence of Bernoulli trials.
For each trial, you check if the outcome is a success (winning at least one of the lotteries) or a failure (winning none). Because each trial is conducted independently, the outcome of one trial does not affect the others. This means that if you win this month, the chances of winning next month remain the same. This independence is key for the trials to fall under the Bernoulli trial category.
Bernoulli trials form the foundational concept that leads into the geometric distribution, which tells us the probability of achieving a success within a number of trials. This concept is crucial for solving the problem and understanding the role of lottery participation as a sequence of Bernoulli trials.
Probability of Winning
In the problem, the probability of winning at least one of the lotteries each month plays a central role. It is important to calculate this correctly, as it serves as the basis for the parameter of the geometric distribution. Here's how this is calculated:
- Let the probability of winning the first lottery be denoted as \( p_1 \), and for the second lottery as \( p_2 \).- The probability of not winning the first lottery is \( 1 - p_1 \), and similarly for the second lottery, it's \( 1 - p_2 \).- To not win any lottery, both must be unsuccessful, which happens with probability \((1 - p_1)(1 - p_2) \).- Therefore, the probability of winning at least one is the complement: \[ p = 1 - (1 - p_1)(1 - p_2) \]This formula allows you to compute the probability of a success in any given month. Understanding this helps predict how many times you'll need to play before hitting a success.
- Let the probability of winning the first lottery be denoted as \( p_1 \), and for the second lottery as \( p_2 \).- The probability of not winning the first lottery is \( 1 - p_1 \), and similarly for the second lottery, it's \( 1 - p_2 \).- To not win any lottery, both must be unsuccessful, which happens with probability \((1 - p_1)(1 - p_2) \).- Therefore, the probability of winning at least one is the complement: \[ p = 1 - (1 - p_1)(1 - p_2) \]This formula allows you to compute the probability of a success in any given month. Understanding this helps predict how many times you'll need to play before hitting a success.
Success Probability
The concept of success probability is central here, determining how often a successful outcome – winning at least one of the lotteries – will occur. The term "success" refers to winning in any lottery round.
Once you calculate the probability of winning at least one lottery as \( p \), this becomes the success probability per trial in a Bernoulli trial setup. This probability \( p \) remains constant across each month you participate.
Having a clear understanding of the success probability helps you to comprehend how these probabilities contribute to defining the geometric distribution that models the number of trials until the first success. In lottery terms, this success probability is a key factor in planning how long you might expect to play before achieving a win.
Once you calculate the probability of winning at least one lottery as \( p \), this becomes the success probability per trial in a Bernoulli trial setup. This probability \( p \) remains constant across each month you participate.
Having a clear understanding of the success probability helps you to comprehend how these probabilities contribute to defining the geometric distribution that models the number of trials until the first success. In lottery terms, this success probability is a key factor in planning how long you might expect to play before achieving a win.
Parameter Calculation
Calculating the parameter of the geometric distribution is crucial to solving the problem. The parameter is essentially the success probability, which we have denoted as \( p \), and it indicates the effectiveness of each attempt.
- First, identify the probability of winning at least one lottery, as derived earlier: \( p = 1 - (1 - p_1)(1 - p_2) \).- This probability is directly used as the parameter for the geometric distribution.- The geometric distribution is defined by how many trials it takes to achieve the first success, thus \( M \), the number of attempts until a win, is geometrically distributed with the parameter \( p \).Understanding parameter calculation helps in visualizing how likelihoods are formed in multi-step problems. This aids in further applications, especially when handling similar probabilistic events. By grasping this concept, you understand not only this mathematical problem but also the underpinnings of probability in daily random events like lotteries.
- First, identify the probability of winning at least one lottery, as derived earlier: \( p = 1 - (1 - p_1)(1 - p_2) \).- This probability is directly used as the parameter for the geometric distribution.- The geometric distribution is defined by how many trials it takes to achieve the first success, thus \( M \), the number of attempts until a win, is geometrically distributed with the parameter \( p \).Understanding parameter calculation helps in visualizing how likelihoods are formed in multi-step problems. This aids in further applications, especially when handling similar probabilistic events. By grasping this concept, you understand not only this mathematical problem but also the underpinnings of probability in daily random events like lotteries.
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