When someone says, “What are the chances of that happening?” what are they asking? Usually when you hear this, it means ‘that’ is very unlikely (as in, “What are the chances that Santa will bring me a pony for Christmas?”). However, this question really has a deeper meaning. Let’s explore what this question really means by focusing on the key word: chances.
When we talk about chances, we are really talking about something called probability. Probability is just how likely an event is to occur. So if something has a high probability, this means it is very likely to happen (as in, what is the probability your mom will make you brush your teeth before bed?). If something has a low probability, it means it is not very likely to happen (as in, what is the probability your mom will let you eat cupcakes for dinner?). Although in reality, it is not always so obvious how likely or unlikely an event is to happen!
When we talk about chances, we are really talking about something called probability. Probability is just how likely an event is to occur. So if something has a high probability, this means it is very likely to happen (as in, what is the probability your mom will make you brush your teeth before bed?). If something has a low probability, it means it is not very likely to happen (as in, what is the probability your mom will let you eat cupcakes for dinner?). Although in reality, it is not always so obvious how likely or unlikely an event is to happen!
So how do we decide how likely or unlikely an event is to happen? More importantly, can we predict how often this event will happen? As it turns out, we can! If we know how many different outcomes there are for a certain event, we can figure out just how likely each one really is.
Here’s an example: take a normal die (this is the singular of dice...so if you have a pair of dice, just use one for this!). A normal die has six sides, each with a different number 1 through 6.
Here’s an example: take a normal die (this is the singular of dice...so if you have a pair of dice, just use one for this!). A normal die has six sides, each with a different number 1 through 6.
If we roll this die, there are six possible numbers we could get, so there are six possible outcomes, and each one is equally likely to happen; one of these outcomes is rolling a number 3. To find the probability of rolling a number 3, we first count how many times the number 3 appears on the die—we know the number 3 appears only one time on this die. Now we count how many possible outcomes there are—we know there are six different possible outcomes, since there are six sides on the die, and each one is equally likely to happen. To find the probability of rolling the number 3, we take the number of times this outcome appears on the die (it appears one time), and divide this number by the total number of possible outcomes (six).
When we divide 1 by 6, we get 0.167 (rounded to three decimal points). What does this mean? Probability may always be expressed as a decimal number between 0 and 1, with 0 meaning the event will never happen, and 1 meaning the event will always happen. This event is closer to 0 than to 1, meaning it is not very likely, but it is not impossible either. Probabilities can also be expressed as a percent (%), with 0 being 0%, and 1 being 100% (just multiply the decimal number by 100 to get the percent chance of this event happening). Rolling a number 3 on our six-sided die therefore has a probability of 16.7% (0.167 X 100 = 16.7%).
An important thing to note about our die is that each of the numbers has an equal chance of coming up on a roll. Since each number only appears one time, we know that each of these numbers has an equal probability of coming up. So the probability of each number is the same as for the number 3, or 16.7%. If we add up the probability of each of the numbers appearing (that is, 16.7% + 16.7% + 16.7% + 16.7% + 16.7% + 16.7%--we have six numbers, all with the same probability, so we add 16.7% six times), we should get a number that is just about 100%. 16.7% + 16.7% + 16.7% + 16.7% + 16.7% + 16.7% = 100.2%; this extra 0.2% is because we rounded to three decimal places—this number is close enough to 100% that we know we calculated correctly. When you calculate probabilities, remember: IF YOU ADD UP THE PROBABILITIES OF ALL THE POSSIBLE EVENTS, THEY SHOULD EQUAL 100%!
An important thing to note about our die is that each of the numbers has an equal chance of coming up on a roll. Since each number only appears one time, we know that each of these numbers has an equal probability of coming up. So the probability of each number is the same as for the number 3, or 16.7%. If we add up the probability of each of the numbers appearing (that is, 16.7% + 16.7% + 16.7% + 16.7% + 16.7% + 16.7%--we have six numbers, all with the same probability, so we add 16.7% six times), we should get a number that is just about 100%. 16.7% + 16.7% + 16.7% + 16.7% + 16.7% + 16.7% = 100.2%; this extra 0.2% is because we rounded to three decimal places—this number is close enough to 100% that we know we calculated correctly. When you calculate probabilities, remember: IF YOU ADD UP THE PROBABILITIES OF ALL THE POSSIBLE EVENTS, THEY SHOULD EQUAL 100%!
TRY THIS!
Here’s what you’ll need:
1. A small bag of Skittles® candies
2. A small paper bag
3. A piece of paper
4. A pen or pencil
Here’s what to do:
1. Pour 10 Skittles out of your bag onto the table. DON’T PICK OUT THE COLORS YOU WANT; JUST MAKE SURE THERE ARE 10 SKITTLES!
Here’s what you’ll need:
1. A small bag of Skittles® candies
2. A small paper bag
3. A piece of paper
4. A pen or pencil
Here’s what to do:
1. Pour 10 Skittles out of your bag onto the table. DON’T PICK OUT THE COLORS YOU WANT; JUST MAKE SURE THERE ARE 10 SKITTLES!
2. Write on your piece of paper the names of the different colors in your sample: Red, Orange, Yellow, Green, and Purple, and divide these with long, vertical lines. Beside the names of the colors, write down the number of the Skittles of that color you have in your sample. For example, if you drew 3 red Skittles, write Red 3.
3. Under the names and numbers for each color, write the number of Skittles of that color, divided by 10 (the total number of Skittles).
4. Place all 10 Skittles in the paper bag.
5. Calculate based on these numbers the probability of each color being drawn from the paper bag at random (that is, without looking or actively choosing the color). For example, for 3 red Skittles, 3/10 should be written, and this works out to 0.3 or 30%. So we know there is a 30% chance of a red Skittle being chosen at random from this sample.
5. Calculate based on these numbers the probability of each color being drawn from the paper bag at random (that is, without looking or actively choosing the color). For example, for 3 red Skittles, 3/10 should be written, and this works out to 0.3 or 30%. So we know there is a 30% chance of a red Skittle being chosen at random from this sample.
6. Add up all the percentages. They should equal 100%, or very close.
7. Reach into the paper bag containing the 10 Skittles, and pull one out. Look at the color, and make a tick mark under that color on your sheet of paper. Put the Skittle back into the bag, and shake the bag to mix the Skittles well.
7. Reach into the paper bag containing the 10 Skittles, and pull one out. Look at the color, and make a tick mark under that color on your sheet of paper. Put the Skittle back into the bag, and shake the bag to mix the Skittles well.
8. Repeat this process 99 times, for a total of 100 tick marks. Be sure to put the Skittle back in the bag each time!
9. Count the number of times each color was randomly selected from the bag. Calculate the percent of the time this color was randomly selected by dividing this number by 100. Compare this number to the percentage you calculated earlier.
What have you found? How close was the percentage you calculated to the number of times out of 100 that each color was chosen? If any of the percentages were very different, why might that be?
WHAT ELSE CAN YOU TRY?
· For this experiment, you pulled a Skittle out of the bag 100 times. That’s a lot, but the more times you repeat your experiment, the more accurate the number will be. Try repeating this experiment 1,000 times! See if the actual numbers get any closer to your predictions!
· You can also try the same experiment with a new sample of Skittles. Pick out 20 different Skittles this time and instead of dividing by 10 to get your probability divide by 20!
· Try this with something different: Use a different candy, or a deck of cards!
Skittles® are a trademark of the Wm. Wrigley Jr. Company.
9. Count the number of times each color was randomly selected from the bag. Calculate the percent of the time this color was randomly selected by dividing this number by 100. Compare this number to the percentage you calculated earlier.
What have you found? How close was the percentage you calculated to the number of times out of 100 that each color was chosen? If any of the percentages were very different, why might that be?
WHAT ELSE CAN YOU TRY?
· For this experiment, you pulled a Skittle out of the bag 100 times. That’s a lot, but the more times you repeat your experiment, the more accurate the number will be. Try repeating this experiment 1,000 times! See if the actual numbers get any closer to your predictions!
· You can also try the same experiment with a new sample of Skittles. Pick out 20 different Skittles this time and instead of dividing by 10 to get your probability divide by 20!
· Try this with something different: Use a different candy, or a deck of cards!
Skittles® are a trademark of the Wm. Wrigley Jr. Company.
