Problem 54
Question
A random number generator selects three numbers from 1 through 10. Find the probability of the event. All three numbers are greater than or equal to \(9 .\)
Step-by-Step Solution
Verified Answer
The probability of the event is 0.008
1Step 1: Determine Total Outcomes
In this scenario, we select one of the ten numbers three times, this gives us \(10^3 = 1000\) potential outcomes.
2Step 2: Determine Successful Outcomes
The successful outcomes occur when we select either 9 or 10 (since these are the only digits greater or equal to 9) three times. This results in \(2 ^ 3 = 8\) successful outcomes.
3Step 3: Calculate Probability
The probability of an event can be found by dividing the number of successful outcomes by the total number of outcomes. Therefore, in this case, it would be the successful outcomes (8) divided by the total outcomes (1000). So, the probability is \( \frac{8}{1000} = 0.008 \)
Key Concepts
Random Number Generator ProbabilityCalculating ProbabilityProbability of Multiple Events
Random Number Generator Probability
In the realm of probability, random number generators (RNG) are tools that produce a sequence of numbers that lack any discernible pattern. When using an RNG to select numbers, as with our textbook exercise, each selection should be independent of the previous one. This independence is crucial for the fundamental principles of probability.
Let's relate this to our problem: An RNG is programmed to pick three numbers between 1 through 10. The key here is that because the generator is random, each number has an equal chance of being selected each time, and the choice of a number in one instance does not influence the next. This uniform distribution means that for each draw, any number from 1 to 10 could come up without bias. In our exercise scenario, only the numbers 9 and 10 are deemed as 'successful' outcomes, which influences the probability calculation.
Let's relate this to our problem: An RNG is programmed to pick three numbers between 1 through 10. The key here is that because the generator is random, each number has an equal chance of being selected each time, and the choice of a number in one instance does not influence the next. This uniform distribution means that for each draw, any number from 1 to 10 could come up without bias. In our exercise scenario, only the numbers 9 and 10 are deemed as 'successful' outcomes, which influences the probability calculation.
Calculating Probability
To calculate the probability of an event, you need to understand two fundamental counts: the total number of outcomes and the number of successful outcomes that satisfy the event's conditions. Probability is then defined as the ratio of successful outcomes to total outcomes.
In our exercise, the total outcomes are determined by raising the number of possible choices (10 numbers) to the power of the number of selections (3 picks), hence we have 103 or 1000 total outcomes. To identify successful outcomes, which are only the instances where all three randomly selected numbers are greater than or equal to 9, we consider only the options 9 and 10, raised to the same power of selections, resulting in 23 or 8 successful outcomes. Finally, the probability is the simple division: \( \frac{8}{1000} \) or 0.008, expressing mathematically that this event is quite unlikely.
In our exercise, the total outcomes are determined by raising the number of possible choices (10 numbers) to the power of the number of selections (3 picks), hence we have 103 or 1000 total outcomes. To identify successful outcomes, which are only the instances where all three randomly selected numbers are greater than or equal to 9, we consider only the options 9 and 10, raised to the same power of selections, resulting in 23 or 8 successful outcomes. Finally, the probability is the simple division: \( \frac{8}{1000} \) or 0.008, expressing mathematically that this event is quite unlikely.
Probability of Multiple Events
When calculating the probability of multiple independent events, the principle of multiplication comes into play. The probability of multiple events occurring simultaneously is the product of their individual probabilities, assuming each event is independent. In our textbook exercise, the event of selecting a number greater than or equal to 9 must happen three times consecutively.
Each selection is an independent event with its own probability, specifically the probability of picking a 9 or 10 on each draw, which is \(\frac{2}{10}\) or 0.2. To find the combined probability of getting a 9 or 10 in all three draws, we multiply the individual probabilities: \(0.2 \times 0.2 \times 0.2\), equating to \(0.008\). This provides us a concise understanding of how multiple probabilities work together to shape the overall likelihood of a sequence of events occurring.
Each selection is an independent event with its own probability, specifically the probability of picking a 9 or 10 on each draw, which is \(\frac{2}{10}\) or 0.2. To find the combined probability of getting a 9 or 10 in all three draws, we multiply the individual probabilities: \(0.2 \times 0.2 \times 0.2\), equating to \(0.008\). This provides us a concise understanding of how multiple probabilities work together to shape the overall likelihood of a sequence of events occurring.
Other exercises in this chapter
Problem 54
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