What Is the Chance You’ll Get a Red M&M?

What Is the Chance You’ll Get a Red M&M?

Probability: Independent vs. Dependent

English: Probability mass function of sum of t...
Probability mass function of sum of two regular dice. (Photo credit: Wikipedia)


  • Paper bag that you cannot see through
  • 10 things of three colors, such as M&Ms, jelly beans, suckers, buttons, tiles
  • Paper
  • Pen or pencil
  • Optional: change the number of colors or items


  • Draw a line across the paper. Above the line, write the name of each color
  • Count the number of items of each color
  • Just below the line, write down the number of items of that color
  • Draw 10 lines below the color names and number them 1 through 10
  • Put all the items in the paper bag
  • Ask your grandchild to pick a color of one of the items and tell you what they think are the chances that they will pick that color out of the bag
  • Their prediction should be the number of items of that color, out of 10
  • That is, if there are 3 reds out of 10 items, there is a 3 out of 10 chance they will pick a red item out of the bag, or 30% (30 out of 100 is the same as 3 out of 10)
  • Have them draw one item out of the bag so that they cannot see the color before they take it out and write a 1 under the color picked
  • Put the item back in the bag
  • Continue 10 times, marking a 1 under whichever color is chosen
  • Add the number of times each color was chosen from the bag
  • Optional: if there are a different number of colors or items, ask your grandchild to predict the chance they will get a certain color by comparing the number of items of that color to the total number of items
  • Optional: Instead of putting the item back in the bag, ask them to predict the chance that they will get the color they choose each time, as a part of the number of that color compared to the total number in the bag. That is, if they started with 5 red items, 3 yellow ones and 2 green ones, the chance they would draw a red the first time was 5 in 10, yellow 3 in 10 and green 2 in 10. If they drew a green the first time, for example, ask them to predict the chance that they’ll get the color they choose the next time. In this case, the chance that they would draw a red would be 5 in 9, the chance they’d draw a yellow 2 in 9 and the chance they’d draw a green again 1 in 9.

What Should Happen?

The number of times a color was picked out of the bag should be close to the percentage of items of that color. That is, if half the items were red, approximately half of the times an item is selected, it should be a red one.

Why Does This Happen?

By putting the item back in the bag every time you pick out a color, you have the same chance of picking out a color as when you started. By repeating the selection ten times, you should get about the same number of picks of each color as you have in the bag. That is, if half are red, you should pick out red about half the time. This is called independent probability. The color you pick is not dependent on the colors you have picked before.

In the optional activity, the probability that you will pick red, for example, is dependent on which colors you have picked before, since this changes the number of items of the different colors left.

Why Is This Useful?

It helps to understand the likelihood, or probability that something will happen to know whether it is independent of other events or dependent on them.

For instance, whenever you throw dice, the chance, or probability that they will come up a certain way is completely independent of whatever number they came up with in the throw before, or the ten throws before. Some people feel that they can get on a lucky streak in dice, for instance, because they seem to get a lot of the throws they want in a row. But, every throw is independent of the throws before because the numbers on the six sides of the dice do not change, so you always have the same chance of getting a certain number or combination of numbers.

Unlike dice, cards, however, vary. The probability of drawing a specific card may be independent or dependent, depending on whether you replace the card in the deck after you draw one. That is, if you draw a card from a deck of 52 cards, you have 4 in 52 chances, or a 1 in 13 chance that it will be an ace. If you replace the card back in the deck, the next time you draw a card, you still have a 4 in 52 chances that it will be an ace.The second draw is independent of whatever you drew the first time, so the chances remain the same. But, if you do not replace the card in the deck, the chances of drawing an ace the second time are now dependent on the first card you drew. So, if you drew an ace the first time, when you had a 4 in 52 chance of drawing one, the second time you will have a 3 in 51 chance of drawing an ace again.

Thanks to proteacher.org for this activity.


Carol Covin, Granny-Guru

Author, “Who Gets to Name Grandma? The Wisdom of Mothers and Grandmothers”


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Comments (13 )

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  1. I learned this with the example of flipping a coin. No matter what there are only two possibilities of the coin flip, either heads or tails.

  2. Jeannette Paladino says:

    I don't know if this is analogous. But buying two lottery tickets instead of one doesn't increase your chance of winning.

    • cheryltherrien says:

      Buying many lottery tickets does not increase my chances of winning. LOL

  3. cheryltherrien says:

    You have that right! Eating them is the best part!

  4. The law of probability is an interesting lesson. Like it is with most, I too learned this same lesson with flipping a coin. For me, I'm with Susan, the best part is in the fun exercise is eating the M&M's. 😀

  5. yearwoodcom says:

    You had me at M&Ms, you can teach me about anything if there is chocolate and peanuts involved.

    I have many faults, but gambling isn't one of them and it's because of the logic you so excellently illustrated here. The minute you understand how odds work you realize what a pointless exercise gambling is. I'd rather give my spare coin to charity. 🙂

    • Geek Girl says:

      I am not a gambler either and this just illustrates why. This is a mighty tasty lesson though!

  6. @patweber says:

    Well then, just have to eat whatever M&Ms come up before I get to the red one. The one I really want. But all of them nevertheless will make getting to the red one fun! And fun is where I am at.

    • Geek Girl says:

      You and me both! 🙂

  7. JeriWB says:

    I also learned this by flipping a coin.

    • Geek Girl says:

      Me too! But M&M's are a much tastier and fun way to learn. 🙂

  8. Julia Reed says:

    That's an interesting question) It's a pity my Mathematics teacher didn't use something of the kind to explain theory of probability to our class.