A particular type of tennis racket comes in a midsize...

A particular type of tennis racket comes in a midsize version and an oversize version. Sixty percent of all customers at a store want the oversize version. (a) Among ten randomly selected customers who want this type of racket, what is the probability that at least six want the oversize version? (b) Among ten randomly selected customers, what is the probability that the number who want the oversize version is within 1 standard deviation of the mean value? (c) The store currently has seven rackets of each version. What is the probability that all of the next ten customers who want this racket can get the version they want from current stock?

A particular type of tennis racket comes in a midsize version and an oversize version. Sixty percent of all customers at a store want the oversize version.
(a) Among ten randomly selected customers who want this type of racket, what is the probability that at least six want the oversize version?
(b) Among ten randomly selected customers, what is the probability that the number who want the oversize version is within 1 standard deviation of the mean value?
(c) The store currently has seven rackets of each version. What is the probability that all of the next ten customers who want this racket can get the version they want from current stock?

Key Concepts

Standard Deviation in the Binomial Distribution

The standard deviation in a binomial distribution quantifies the spread or variability of the number of successes across trials. It is calculated using the formula sqrt(n * p * (1-p)), where n is the number of trials and p is the probability of success. This concept is key when assessing how much deviation from the mean is expected in a binomial setting.

Inventory/Stock-Out Analysis

Inventory or stock-out analysis in probabilistic contexts involves determining the probability that the available stock can meet the demand, which is modeled through probabilistic outcomes such as those given by a binomial distribution. This analysis is crucial for decision-making in logistics and supply chain management, ensuring that the supply can successfully cover the random demand within predefined limits.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has two outcomes (success or failure) with the same probability of success. It is essential for computing probabilities in scenarios where a fixed number of customers, items, or events occur, each with an independent chance of a particular outcome.

Cumulative Probability

Cumulative probability involves summing the individual probabilities of a random variable falling within a range of values. In the context of binomial distributions, it is used to compute the probability that the number of successes is at least or within a certain interval by adding up the probabilities for each potential count in that range.

Transcript

00:01 In this question, we're told that a tennis racket comes in an oversized version, and 60 % of all customers want the oversized version.

00:11 And then we're asked some probability questions about a sample of 10 customers.

00:17 So we can say that we have p is equal to 0 .6, and n equals 10.

00:30 And let's let x be the number of successes in 10 customers.

00:34 And this is a binomial random variable based on a sample size of 10 and a probability of success of 0 .6.

00:53 So a basically asks what is the probability out of 10 people of at least six successes? we can rewrite this as 1 minus the probability of at most 5 successes.

01:32 So this is 1 minus 0 .367.

01:41 Comes out to the probability of 0 .633.

01:47 And now b asks, among the 10 randomly selected customers, what is the probability that we have a number of customers that are within one standard deviation of the mean value? so we must calculate the mean and the standard deviation for this distribution.

02:08 So the mean for binomial distribution is n times p, and that is 10 times 0 .0.

02:17 6, which equals 6, and the standard deviation is equal to the square root of n times p times q, and that comes out to 1 .549.

02:57 So to define the range that's within one standard deviation of the mean, it's looking for the range that is the mean plus or minus one standard deviation.

03:15 And so it's 6 plus or minus 1 .549, and that gives us a range.

03:28 Of 4 .45 to 7 .55.

03:38 So what is the probability of getting between 4 .45 and 7 .55 successes? now in a binomial distribution, you can only have an integer number of successes.

03:51 So we're really looking for the probability that x is between, we'll say this is 5 and this is 7.

04:03 Because if x is 4 or less, or if it's 8 or more, it's 8 or more, it's a probability, it's more than a standard deviation away.

04:10 So we're looking for the probability that x is between 5 and 7, inclusive...

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