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Answer :
Sure! Here is the step-by-step solution for the given problem:
We are asked to find the probability of winning no more than 8 times out of 10 plays of an arcade game.
1. Understand the type of distribution: Since the game can either be a win or a loss and each play is an independent event, this scenario follows a binomial distribution. In a binomial distribution, we denote:
- [tex]\( n \)[/tex] as the number of trials (in this case, 10),
- [tex]\( p \)[/tex] as the probability of success (winning the game), given as 0.632,
- [tex]\( k \)[/tex] as the number of successful trials we are interested in (no more than 8 in this case).
2. Set up the problem: We need to calculate the cumulative probability of winning 0 to 8 times out of 10, which is represented as [tex]\( P(X \leq 8) \)[/tex].
3. Calculate the cumulative probability:
To find this probability, we use the cumulative distribution function (CDF) for the binomial distribution. The CDF sums up the probabilities of all outcomes from [tex]\( k = 0 \)[/tex] to [tex]\( k = 8 \)[/tex].
4. Result: Using the cumulative distribution function for the binomial distribution with [tex]\( n = 10 \)[/tex], [tex]\( p = 0.632 \)[/tex], and [tex]\( k \leq 8 \)[/tex], the probability [tex]\( P(X \leq 8) \)[/tex] is:
[tex]\[ P(X \leq 8) = 0.9306 \][/tex]
So, the probability of winning no more than 8 times out of 10 plays of the arcade game is approximately [tex]\( 0.9306 \)[/tex] or [tex]\( 93.06\% \)[/tex].
I hope this explanation helps you understand how to approach and solve binomial probability problems! If you have any more questions, feel free to ask!
We are asked to find the probability of winning no more than 8 times out of 10 plays of an arcade game.
1. Understand the type of distribution: Since the game can either be a win or a loss and each play is an independent event, this scenario follows a binomial distribution. In a binomial distribution, we denote:
- [tex]\( n \)[/tex] as the number of trials (in this case, 10),
- [tex]\( p \)[/tex] as the probability of success (winning the game), given as 0.632,
- [tex]\( k \)[/tex] as the number of successful trials we are interested in (no more than 8 in this case).
2. Set up the problem: We need to calculate the cumulative probability of winning 0 to 8 times out of 10, which is represented as [tex]\( P(X \leq 8) \)[/tex].
3. Calculate the cumulative probability:
To find this probability, we use the cumulative distribution function (CDF) for the binomial distribution. The CDF sums up the probabilities of all outcomes from [tex]\( k = 0 \)[/tex] to [tex]\( k = 8 \)[/tex].
4. Result: Using the cumulative distribution function for the binomial distribution with [tex]\( n = 10 \)[/tex], [tex]\( p = 0.632 \)[/tex], and [tex]\( k \leq 8 \)[/tex], the probability [tex]\( P(X \leq 8) \)[/tex] is:
[tex]\[ P(X \leq 8) = 0.9306 \][/tex]
So, the probability of winning no more than 8 times out of 10 plays of the arcade game is approximately [tex]\( 0.9306 \)[/tex] or [tex]\( 93.06\% \)[/tex].
I hope this explanation helps you understand how to approach and solve binomial probability problems! If you have any more questions, feel free to ask!
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