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Sagot :
Certainly! Let's break down the problem and find the probability step-by-step.
### Problem Statement
We have a six-sided die, and it is rolled ten times. We are asked to find the probability that it will show an even number at most eight times.
### Explanation and Calculation
1. Definition of Success and Parameters:
- A six-sided die has numbers 1, 2, 3, 4, 5, and 6.
- The even numbers are 2, 4, and 6.
- The probability of rolling an even number (success) in a single roll is [tex]\( \frac{3}{6} = \frac{1}{2} \)[/tex].
2. Number of Trials:
- We roll the die 10 times. So, the number of trials [tex]\( n = 10 \)[/tex].
3. Random Variable:
- Let [tex]\( X \)[/tex] be the random variable that represents the number of times the die shows an even number.
- [tex]\( X \)[/tex] follows a Binomial distribution [tex]\( X \sim \text{Binomial}(n = 10, p = \frac{1}{2}) \)[/tex].
4. Objective:
- We need to find [tex]\( P(X \leq 8) \)[/tex], i.e., the probability that [tex]\( X \)[/tex] is at most 8.
### Binomial Distribution Calculation
- The probability mass function (PMF) of a Binomial distribution is given by:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
- The cumulative distribution function (CDF) up to [tex]\( k = 8 \)[/tex] gives us [tex]\( P(X \leq 8) \)[/tex].
### Using the Result
Given the problem conditions and the parameters specified:
- Number of trials [tex]\( n = 10 \)[/tex]
- Probability of success [tex]\( p = \frac{1}{2} \)[/tex]
The probability that the die will show an even number at most eight times is calculated as approximately [tex]\( 0.9892578125 \)[/tex].
### Finding the Exact Match
Now we need to match this probability with one of the given answer choices:
- Option A: [tex]\( \frac{1013}{1024} \)[/tex]
[tex]\[ \frac{1013}{1024} \approx 0.9892578125 \][/tex]
- Option B: [tex]\( \frac{1}{2} \)[/tex]
[tex]\[ \frac{1}{2} = 0.5 \][/tex]
- Option C: [tex]\( \frac{37}{250} \)[/tex]
[tex]\[ \frac{37}{250} = 0.148 \][/tex]
- Option D: [tex]\( \frac{247}{250} \)[/tex]
[tex]\[ \frac{247}{250} = 0.988 \][/tex]
Clearly, Option A [tex]\( \frac{1013}{1024} \)[/tex] is the exact match to the calculated probability.
### Conclusion
Hence, the probability that the die will show an even number at most eight times is:
[tex]\[ \boxed{\frac{1013}{1024}} \][/tex]
This corresponds to Option A.
### Problem Statement
We have a six-sided die, and it is rolled ten times. We are asked to find the probability that it will show an even number at most eight times.
### Explanation and Calculation
1. Definition of Success and Parameters:
- A six-sided die has numbers 1, 2, 3, 4, 5, and 6.
- The even numbers are 2, 4, and 6.
- The probability of rolling an even number (success) in a single roll is [tex]\( \frac{3}{6} = \frac{1}{2} \)[/tex].
2. Number of Trials:
- We roll the die 10 times. So, the number of trials [tex]\( n = 10 \)[/tex].
3. Random Variable:
- Let [tex]\( X \)[/tex] be the random variable that represents the number of times the die shows an even number.
- [tex]\( X \)[/tex] follows a Binomial distribution [tex]\( X \sim \text{Binomial}(n = 10, p = \frac{1}{2}) \)[/tex].
4. Objective:
- We need to find [tex]\( P(X \leq 8) \)[/tex], i.e., the probability that [tex]\( X \)[/tex] is at most 8.
### Binomial Distribution Calculation
- The probability mass function (PMF) of a Binomial distribution is given by:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
- The cumulative distribution function (CDF) up to [tex]\( k = 8 \)[/tex] gives us [tex]\( P(X \leq 8) \)[/tex].
### Using the Result
Given the problem conditions and the parameters specified:
- Number of trials [tex]\( n = 10 \)[/tex]
- Probability of success [tex]\( p = \frac{1}{2} \)[/tex]
The probability that the die will show an even number at most eight times is calculated as approximately [tex]\( 0.9892578125 \)[/tex].
### Finding the Exact Match
Now we need to match this probability with one of the given answer choices:
- Option A: [tex]\( \frac{1013}{1024} \)[/tex]
[tex]\[ \frac{1013}{1024} \approx 0.9892578125 \][/tex]
- Option B: [tex]\( \frac{1}{2} \)[/tex]
[tex]\[ \frac{1}{2} = 0.5 \][/tex]
- Option C: [tex]\( \frac{37}{250} \)[/tex]
[tex]\[ \frac{37}{250} = 0.148 \][/tex]
- Option D: [tex]\( \frac{247}{250} \)[/tex]
[tex]\[ \frac{247}{250} = 0.988 \][/tex]
Clearly, Option A [tex]\( \frac{1013}{1024} \)[/tex] is the exact match to the calculated probability.
### Conclusion
Hence, the probability that the die will show an even number at most eight times is:
[tex]\[ \boxed{\frac{1013}{1024}} \][/tex]
This corresponds to Option A.
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