A manufacturer of game controllers is concerned that their controller may be difficult for​ left-handed users. They set out to find lefties to test. Suppose that about 85​% of the population is​ right-handed. They select a sample of twelve customers at random in their stores. Complete parts a through e below.
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Part 1
​a) Find the mean and standard deviation of the number of​ right-handers in the group.
The mean number of righties is 
  
10.20.
The standard deviation would be 
  
1.24. 
​(Type integers or decimals rounded to two decimal places as​ needed.)
Part 2
​b) What is the probability that they are not all​ right-handed?
The probability that they are not all righthanded is 
  
0.858.
​(Round to three decimal places as​ needed.)
Part 3
​c) What is the probability that there are no more than 10 ​righties?
The probability that there are no more than 10 righties is 
  
enter your response here.
​(Round to three decimal places as​ needed.)

We are working with a binomial distribution problem where:

- \( n = 12 \) (sample size),
- \( p = 0.85 \) (probability of a customer being right-handed),
- \( q = 1 - p = 0.15 \) (probability of a customer being left-handed).

### Part (c): Finding the probability that there are no more than 10 right-handers

To calculate the probability that there are no more than 10 right-handers, we need to find:

\[
P(X \leq 10)
\]

Where \( X \) is the number of right-handed customers in the sample, which follows a binomial distribution:

\[
P(X = k) = \binom{n}{k} p^k q^{n-k}
\]

We can compute the cumulative probability for \( P(X \leq 10) \) by summing the probabilities for all possible values of \( X \) from 0 to 10:

\[
P(X \leq 10) = P(X = 0) + P(X = 1) + \dots + P(X = 10)
\]

Let me calculate that for you.The probability that there are no more than 10 right-handers in the group is approximately \( 0.557 \) (rounded to three decimal places).

Would you like further details or explanations? 

Here are 5 related questions:
1. What is the probability that there are exactly 10 right-handers?
2. How would the probability change if the sample size increased to 20?
3. Can you calculate the probability that there are fewer than 8 right-handed customers?
4. What is the probability of having exactly 12 right-handed people?
5. How does the standard deviation affect the interpretation of this problem?

**Tip:** The binomial distribution is useful when you have a fixed number of independent trials, each with two possible outcomes.

A manufacturer of game controllers is concerned that their controller may be difficult for left-handed users. They select a sample of twelve customers at random in their stores. The mean number of righties is 10.20, and the standard deviation is 1.24. Calculate the probability that there are no more than 10 right-handers.

Explore how to calculate the probability that no more than 10 out of 12 randomly selected customers are right-handed. Learn about binomial distribution, expected value, and variance with a step-by-step solution. This problem is perfect for students in Grades 10-12 studying probability theory.

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