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Answer :
Final answer:
The problem involves determining the probability under a binomial distribution of having up to 10 successes (wins) in 21 trials (games). This requires finding and summing the probabilities of having exactly k successes for each possible value of k from 0 to 10, which can be computed using a binomial distribution formula or a binomial distribution calculator.
Explanation:
This problem involves the concept of a binomial distribution in probability. A binomial distribution describes the number of successes in a specified number of independent Bernoulli trials with the same probability of success. In this context, each game is a Bernoulli trial where you either win (success) or lose (failure).
In a binomial distribution, the probability of getting exactly k successes in n trials is given by the formula:
[tex]P(X=k) = [nCk] * (p^k) * (1-p)^(^n^-^k^)[/tex]
where:
- [nCk] denotes the binomial coefficient, which is the number of combinations of n items taken k at a time.
- p is the probability of success on an individual trial.
- 1-p is the probability of failure on an individual trial.
The formula calculates the probability for one specific value of k (the number of successes). However, the problem asks for the probability of winning no more than 10 times. This means we have to calculate and add up the probabilities for k = 0, 1, 2, ..., 10.
Note: This problem can be solved directly using a binomial distribution calculator.
Please note that these calculations can be complex and labor-intensive without a calculator that's capable of determining binomial probabilities.
Learn more about Binomial Distribution here:
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Final answer:
The probability of winning no more than 10 times in 21 plays of an arcade game with a win probability of 0.612 can be calculated using the binomial probability formula. This is a complex computation best handled with statistical software or a scientific calculator.
Explanation:
This question is about calculating the probability of winning a certain amount of times in a game, specifically an arcade game where the probability of winning is 0.612. The student would like to know about the odds of winning no more than 10 times out of 21 plays.
To calculate this, we need to use the binomial probability formula, which is P(x) = [n! / (x!(n-x)!)] * (p^x) * (q^(n-x)), where n represents the number of trials, x represents the desired number of successes, p is the probability of success on a single trial, and q is the probability of failure on a single trial.
In this particular scenario, you are asked to calculate the probability of winning no more than 10 times out of 21, meaning x can take values from 0 to 10, and the probabilities for each of these should be added together. The probability of success (winning) p = 0.612, and the probability of failure 1-p = 0.388.
This calculation requires a significant amount of computation and might best be achieved using statistical software or a scientific calculator with a built-in binomial probability function.
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