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Answer :
To solve this problem, we need to determine the probability of winning no more than 14 times out of 28 attempts in a game where each attempt has a winning probability of 0.648. This scenario is a classic example of a binomial probability problem.
The key elements of a binomial distribution are:
Number of trials (n): 28
Probability of success on each trial (p): 0.648
Probability of failure on each trial (q): 1 - p = 0.352
Number of successful trials (x): a variable that we adjust to meet the requirement "no more than 14 times"
To find the probability of winning no more than 14 times, we have to sum the binomial probabilities for all numbers of success from 0 to 14.
Mathematically, the probability of exactly k successes (in this case, wins) in n trials is given by the binomial probability formula:
[tex]P(X = k) = \binom{n}{k} p^k q^{n-k}[/tex]
where [tex]\binom{n}{k}[/tex] is the binomial coefficient, calculated as:
[tex]\binom{n}{k} = \frac{n!}{k!(n-k)!}[/tex]
Now, we compute the total probability of winning no more than 14 times as:
[tex]P(X \leq 14) = \sum_{k=0}^{14} P(X = k)[/tex]
This involves calculating [tex]P(X = k)[/tex] for each [tex]k[/tex] from 0 to 14 and summing these probabilities.
Using technology, such as a calculator or software with statistical functions, can significantly simplify this process. However, here is the conceptual approach using the binomial formula:
Calculate [tex]P(X = k)[/tex] for [tex]k = 0, 1, 2, \ldots, 14[/tex].
Sum these probabilities to get [tex]P(X \leq 14)[/tex].
On most calculators or statistical software, you can use the cumulative distribution function (cdf) of a binomial distribution to directly find this probability:
[tex]P(X \leq 14) = \text{binom.cdf}(14, 28, 0.648)[/tex]
After performing these calculations, you should find:
[tex]P(X \leq 14) \approx 0.094[/tex] (rounded to three decimal places).
This means there is approximately a 9.4% probability of winning no more than 14 times when playing the arcade game 28 times.
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