You are the meat products manager for Gigantic Foods, a...

You are the meat products manager for Gigantic Foods, a large retail supermarket food distributor who is studying the characteristics of its whole chicken product mix. Chickens are purchased from both Free Range Farms and Big Foods Ltd. Free Range Farms produces chickens that are fed with natural grains and grubs in open feeding areas. In their product mix, $10 \%$ of the processed chickens weigh less than 3 pounds. Big Foods Ltd. produces chickens in cages using enriched food grains for rapid growth. They note that $20 \%$ of their processed chickens weigh less than three poounds. Gigantic Foods purchases $40 \%$ of its chickens from Free Range Farms and mixes the products together with no identification of the supplier. Suppose you purchase a chicken that weighs more than three pounds. What is the probability the chicken came from Free Range Farms? If you purchase 5 chickens, what is the probability that at least 3 came from Free Range Farms?

Key Concepts

Bayes' Theorem

Bayes’ Theorem provides a way to update the probability for a hypothesis based on new evidence. It is used to reverse conditional probabilities, meaning it can compute the probability that an event originates from a particular source given that another event (such as a measurement or trait) has occurred.

Law of Total Probability

The Law of Total Probability is used to break down the probability of an event into the sum of probabilities over all mutually exclusive ways that event can occur. It is particularly useful when a problem involves a mixture of different sources or populations, ensuring that all pathways contributing to the outcome are accounted for.

Binomial Distribution

The Binomial Distribution models the number of successes in a sequence of independent trials with two outcomes (success or failure). It applies when determining the probability of a specific number of events (such as items coming from a certain group) in a fixed number of trials, especially when each trial has the same probability of success.

Mixture Models

Mixture Models are frameworks for analyzing data that arise from several distinct groups or sources. They account for different probability distributions within a single overall distribution, making them useful for scenarios where items are combined from multiple sources with varying characteristics.

Transcript

00:01 So we're dealing with chickens, and we have this gigantic food company that purchases 40 % of its chickens come from the free range farm, and therefore 60 % come from the big food farm.

00:17 And we know that from the free range farm, a 10 % are less than 3 pounds, and therefore 90 % are at least 3 pounds.

00:28 And for the big food, 20 % of the chickens are less than 3 pounds, and therefore 80 % are at least 3 pounds.

00:41 And we want to find the probability, this is kind of a, we want to find the probability that given that you've got a chicken that weighs under 3 pounds, what's the likelihood that it came from the free range farm? so we know we want to find the probability of free range and weighs less than three pounds, divided by the probability of weighing less than three pounds.

01:11 And the less than three pounds has two ways of happening, either this way or this way.

01:17 And so this times this is the 0 .4 times 0 .1 plus 0 .6 times 0 .2.

01:25 And so that probability comes out to be the 0 .4 times 0 .1 plus 0 .6 times 0 .2 comes out to be 0 .16.

01:40 And the part of that comes, that is the intersection or it comes from the free range, is this 0 .04.

01:48 So we would have a 1 4th chance or a 25 % chance that it came from the free range farm.

01:55 Now, your next question, i'll call it part b, says that you get five of those chickens.

02:03 And we know that the probability that it comes from the free range farm is a one -fourth chance.

02:10 And we want to know what's the probability that at least three of them came from, probability that at least three of them came from the free range farm.

02:23 And so this is equivalent to, this is a binomial setting.

02:29 And we know this is the chance that they come from the free range farm.

02:33 And so you have a three -fourths chance of coming from the other farm.

02:37 And we would have three, four, and five...

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