# Probability of Getting Blue M&M from Mini Bag of 20

• lopko
In summary, the question asks for the probability that the 20th M&M in a mini bag of 20 M&Ms is blue, given that the first 19 were all blue. According to the given distribution from the company, the probability of getting a blue M&M is 24%. The events of drawing each M&M are assumed to be independent, making the probability the same for each draw. Therefore, the probability of the 20th M&M being blue is also 24%.
lopko

## Homework Statement

Lets say you have a mini bag of 20 M&Ms and find the first 19 are all blue. What is the probability that the 20th is also blue?

According to the company the distribution is 24% blue, 14% brown, 16% green, 20% orange, 13% red, 14% yellow

Thanks!

## Homework Equations

I don't think this is relevant

## The Attempt at a Solution

Would it just be the initial probability of getting a blue M&M (.24 according to the company), or does the fact that ALL of them in the bag are blue have an effect on the probability? I am assuming that the picking of each M&M are independent events, thus making the probability the same... but I am not sure if that's correct

lopko said:

## Homework Statement

Lets say you have a mini bag of 20 M&Ms and find the first 19 are all blue. What is the probability that the 20th is also blue?

According to the company the distribution is 24% blue, 14% brown, 16% green, 20% orange, 13% red, 14% yellow

Thanks!

## Homework Equations

I don't think this is relevant

## The Attempt at a Solution

Would it just be the initial probability of getting a blue M&M (.24 according to the company), or does the fact that ALL of them in the bag are blue have an effect on the probability? I am assuming that the picking of each M&M are independent events, thus making the probability the same... but I am not sure if that's correct

That would be correct.

Thanks Dick. What's the reasoning behind it?

lopko said:
Thanks Dick. What's the reasoning behind it?

They gave you the distribution of colors. And there's no reason not to assume the events are independent. The colors you've already seen don't affect the probability of the color of the next one. Of course, after pulling out 20 blues you might suspect the color distribution you've been given is wrong. But that really has nothing to do with this problem.

lopko said:

## Homework Statement

Lets say you have a mini bag of 20 M&Ms and find the first 19 are all blue. What is the probability that the 20th is also blue?

According to the company the distribution is 24% blue, 14% brown, 16% green, 20% orange, 13% red, 14% yellow

Thanks!

## Homework Equations

I don't think this is relevant

## The Attempt at a Solution

Would it just be the initial probability of getting a blue M&M (.24 according to the company), or does the fact that ALL of them in the bag are blue have an effect on the probability? I am assuming that the picking of each M&M are independent events, thus making the probability the same... but I am not sure if that's correct

You need to be careful. The point is that the contents of the bag were fixed at the M&M factory, so the question is about the number of blues that were put into the bag at the plant. The probability 0.24 governs what was put into the bag, not necessarily what will be taken out be a customer. In particular, you need to worry about whether the successive colors drawn out of the bag by a customer are truly independent.

That's what I thought but I was involved in an argument here. Thanks.
I am confused about the question below. It is a rider to the one above.I believe it is a combination since order is not spoken of, but I am not sure.

If you are given 10 Brown M&Ms, 5 Yellow,16 Green, 7 Red, 11 orange and 6 blue, What would be FORMULA for the probability of obtaining ANY given combination of colors? Please define all variables used.
Sorry for being such an Oliver Twist.

Dick said:
They gave you the distribution of colors. And there's no reason not to assume the events are independent. The colors you've already seen don't affect the probability of the color of the next one. Of course, after pulling out 20 blues you might suspect the color distribution you've been given is wrong. But that really has nothing to do with this problem.

I think the question wants P{X = 20|X >= 19} in a binomial distribution X~Bin(20,0.24). That conditional probability is definitely not equal to 0.24. Further, P{X = k+1|X >=k} depends on k, so the results are not independent!

Last edited:
Ray Vickson said:
P{X >= k+1|X >=k} depends on k, so the results are not independent!
Ray, I don't think that's relevant. That's where you're looking at the total number of blues in the bag, not the first so many drawn at random. The logic in the OP looks right to me. Pulling the Mth M&M from the Bth bag of N M&Ms is equivalent to pulling the (B-1)*N+Mth from the production line.

## What is the probability of getting a blue M&M from a mini bag of 20?

The probability of getting a blue M&M from a mini bag of 20 is 1 in 5, or 20%. This is because there are five different colors in a bag of M&Ms (red, orange, yellow, green, and blue), and each color has an equal chance of being selected.

## What factors can affect the probability of getting a blue M&M?

The only factor that can affect the probability of getting a blue M&M is if the bag is not evenly mixed and contains more or less blue M&Ms than the other colors. However, assuming the bag is evenly mixed, the probability remains the same for each M&M.

## What is the total number of blue M&Ms in a mini bag of 20?

Since the probability of getting a blue M&M is 20%, there would be an expected 4 blue M&Ms in a mini bag of 20. However, this is not a guarantee as the number could be slightly higher or lower due to chance.

## What is the probability of not getting a blue M&M from a mini bag of 20?

The probability of not getting a blue M&M from a mini bag of 20 is 4 in 5, or 80%. This can be calculated by subtracting the probability of getting a blue M&M (1 in 5) from 1 (100%).

## How does the probability of getting a blue M&M change if there are multiple bags with 20 M&Ms each?

The probability of getting a blue M&M does not change with multiple bags. Each bag of 20 M&Ms has the same probability of 20% for getting a blue M&M, regardless of how many bags there are.

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